Rule 30 inversion The total numbers of sequences for t from 1 to 15 not yielding stripes of heights 1 and 2 are respectively {1, 2, 2, 3, 3, 6, 6, 10, 16, 31, 52, 99, 165, 260} {2, 5, 8, 14, 23, 40, 66, 111, 182, 316, 540, 921, 1530, 2543, 4122} The sideways evolution of rule 30 discussed on page 601 implies that if one fills cells from the left rather than the right then some sequence of length t + 1 will always yield any given stripe of height t . If the evolution of rule 30 can be set up as on page 704 to emulate any Boolean function then the problem considered here is immediately equivalent to satisfiability.

Starting in the 1960s text editors like TECO and ed used sequential substitution system rules, as have string-processing languages such as SNOBOL and perl. Mathematica uses an analog of sequential substitution system rules to transform general symbolic expressions. The fact that new rules can be added to a sequential substitution system incrementally without changing its basic structure has made such systems popular in studies of adaptive programming.

[State networks for] shift rules The pictures below show networks obtained with rule 170, which just shifts every configuration one position to the left at each step. With any such shift rule, all states lie on cycles, and the lengths of these cycles are the divisors of the size n .

But while we have discussed a whole range of different kinds of underlying rules, we have for the most part considered only the simplest possible initial conditions—so that for example we have usually started with just a single black cell. … The picture at the top of the next page shows as a simple first example a cellular automaton which starts from a typical random initial condition, then evolves down the page according to the very simple rule that a cell becomes black if either of its neighbors are black.

Any program can at some level be thought of as consisting of a set of rules that specify what it should do at each step. There are many possible ways to set up these rules—and indeed we will study quite a few of them in the course of this book.

The pictures below now compare what happens in the rule 30 cellular automaton from page 27 if one starts from random initial conditions and from initial conditions involving just a single black cell. Comparison of the patterns produced by the rule 30 cellular automaton starting from random initial conditions and from simple initial conditions involving just a single black cell.

Indeed, in the particular case of systems such as random walks, the Central Limit Theorem suggested over two centuries ago ensures that for a very wide range of underlying microscopic rules, the same continuous so-called Gaussian distribution will always be obtained. … A demonstration of the fact that for a wide range of underlying rules for each step in a random walk, the overall distribution obtained always has the same continuous form.

Nesting can be defined by thinking in terms of splitting into smaller and smaller elements according to some fixed rule. … Part of it is that the same basic rules must apply regardless of physical scale.

One can capture this basic effect by having a cellular automaton with rules in which cells become black if they have exactly one black neighbor, but stay white whenever they have more than one black neighbor. … And despite the simplicity of its underlying rules, what one sees is that the patterns it produces are strikingly similar to those seen in real snowflakes.

The rule on the left shows the amount of material added at each stage at different points around the opening; the line from the center indicates the progressive lateral displacement of the opening. … All shells produced by adding material according to fixed rules of the kind shown here have the property that throughout their growth they maintain the same overall shape.