2-neighbor [class 4] rules Among 3-color 2-neighbor rules class 4 behavior seems to be comparatively rare; the picture below shows an example with rule number 2144.

[No text on this page] Persistent structures found in rule 110.

[No text on this page] Examples of network evolution according to the same basic underlying rules as on page 511 , but now with all possible clusters of nodes that do not overlap being replaced at each step.

For we just used a simple cellular automaton rule, and just started from a simple initial condition containing a single black cell. … The next two pages [ 29 , 30 ] show progressively more steps in the evolution of the rule 30 cellular automaton from the previous page . … Yet given the simplicity of the underlying rule, one would expect vastly more regularities.

The basic form of the rule is just the same as before. But now the specific rule used—that I call rule 110—takes the new color of a cell to be black in every case except when the previous colors of the cell and its two neighbors were all the same, or when the left neighbor was black and the cell and its right neighbor were both white. The pattern obtained with this rule shows a remarkable mixture of regularity and irregularity.

And no doubt it is from such simple rules of growth that most such overall branching patterns come. … Can they also be thought of as consequences of simple underlying rules of growth? … So given this diversity one might at first suppose that no single kind of underlying rule could be responsible for what is seen.

In rule 22, both edges expand about 0.7660 cells on average per step. The motion of the right-hand edge in rule 30 can be understood by noting that with this rule the color of a particular cell will always change if the color of the cell to its left is changed on the previous step (see page 601 ). … (For rule 45, the left-hand edge of the difference pattern moves about 0.1724 cells per step; for rule 54 both edges move about 0.553 cells per step.)

As discussed in the main text (see also page 1155 ) one can think of axioms as giving rules for transforming symbolic expressions—much like rules in Mathematica. … In predicate logic the tab at the top specifies how to construct rules (which in this case are often called rules of inference, as discussed on page 1155 ). x_ ∧ y_  x_ is the modus ponens or detachment rule (see page 1155 ). x_  ∀ y_ x_ is the generalization rule. x_  x_ ∧ # & is applied to the axioms given to get a list of rules. … But in predicate logic rules can be applied only to whole expressions, always in effect using Replace[expr, rules] .

[No text on this page] The effect of small changes in initial conditions in the rule 110 class 4 cellular automaton.

Rule 218 [with simple initial conditions] If pairs of adjacent black cells appear anywhere in its initial conditions this class 2 rule gives uniform black, but if none do it gives a rule 90 nested pattern.