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The World of Simple Programs | More Cellular Automata


rules 0 and 128 all the cells become white, while in rule 255 all of them become black. There are also rules such as 7 and 127 in which all cells alternate between black and white on successive steps.

But among the rules shown on the last few pages [54, 55, 56] the single most common kind of behavior is one in which a pattern consisting of a single cell or a small group of cells persists. Sometimes this pattern remains stationary, as in rules 4 and 123. But in other cases, such as rules 2 and 103, it moves to the left or right.

It turns out that the basic structure of the cellular automata discussed here implies that the maximum speed of any such motion must be one cell per step. And in many rules, this maximum speed is achieved--although in rules such as 3 and 103 the average speed is instead only half a cell per step.

In about two-thirds of all the cellular automata shown on the last few pages [54, 55, 56], the patterns produced remain of a fixed size. But in about one-third of cases, the patterns instead grow forever. Of such growing patterns, the simplest kind are purely repetitive ones, such as those seen in rules 50 and 109. But while repetitive patterns are by a small margin the most common kind, about 14% of all the cellular automata shown yield more complicated kinds of patterns.

The most common of these are nested patterns, like those on the next page. And it turns out that although 24 rules in all yield such nested patterns, there are only three fundamentally different forms that occur. The simplest and by far the most common is the one exemplified by rules 22 and 60. But as the pictures on the next page show, other nested forms are also possible. (In the case of rule 225, the width of the overall pattern does not grow at a fixed rate, but instead is on average proportional to the square root of the number of steps.)

Captions on this page:

The behavior of all 256 possible cellular automata with rules involving two colors and nearest neighbors. In each case, thirty steps of evolution are shown, starting from a single black cell. Note that some of the rules are related just by interchange of left and right or black and white (e.g. rules 2 and 16 or rules 126 and 129). There are 88 fundamentally inequivalent such elementary rules.


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Notes related to this page:

* Rule equivalences [for cellular automata]
* Special [cellular automaton] rules
* Rule expressions [for cellular automata]
* Rule orderings [for cellular automata]
* Rule 225
* Rule 22
* Implementing cellular automata
* Built-in cellular automaton function
* History of cellular automata
* All notes for this section
* Downloadable programs for this page
* Downloadable images
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From Stephen Wolfram: A New Kind of Science [citation] Previous page-----Next page