Its rule is extremely simple—involving just nearest neighbors and two colors of cells. … So are there also structures in rule 110 that exhibit unbounded growth? … So how do the various structures in rule 110 interact?

[No text on this page] Examples of the evolution of two-dimensional cellular automata with various totalistic rules starting from random initial conditions. The rules involve a cell and its four immediate neighbors. Each successive base 2 digit in the code number for the rule gives the outcome when the total of the cell and its four neighbors runs from 5 down to 0.

[No text on this page] Examples of three-color totalistic rules that yield patterns which grow forever but have a fundamentally repetitive structure. Examples of three-color totalistic rules which yield nested patterns. In most cases, these patterns have an overall form that is similar to what was found with two-color rules.

One possibility, illustrated in the pictures below, is to have a system that evolves in time according to explicit rules, but for these rules to have built into them a symmetry between space and time. … Each picture can be generated by starting from initial conditions at the top, and then just evolving down the page repeatedly applying the cellular automaton rule. The particular rules shown are reversible second-order ones with numbers 90R and 150R.

But modifying the rule just slightly one can immediately get a different pattern. … TEST A representation of the rule for the cellular automaton shown above. … In the numbering scheme described in Chapter 3 , this is cellular automaton rule 254.

In the top rule, the new color of a particular cell is found simply by looking at that cell and its immediate neighbors above and to the right. … The bottom rule above is exactly the same as was shown on page 336 . Whichever color was initially more common again eventually dominates, though with this rule it takes somewhat longer for this to occur.

If theoretical science is to be possible at all, then at some level the systems it studies must follow definite rules. Yet in the past throughout the exact sciences it has usually been assumed that these rules must be ones based on traditional mathematics. … Earlier in history it might have been difficult to imagine what more general types of rules could be like.

The reason for this is that the basic rules we used specify that every single element should be replaced by at least one new element. … Rules of this kind cannot readily be interpreted in terms of simple subdivision of one element into several. … Note that on every step the rightmost element is always dropped, since no rule is given for how to replace it.

In the rather simple case of rule (a) the results turn out to be independent of the updating scheme that was used. But for rules (b) and (c), different schemes in general yield different causal networks. So what kinds of underlying replacement rules lead to causal networks that are independent of how the rules are applied?

Although their basic rules are more complicated, the cellular automata shown here do not seem to have fundamentally more complicated behavior than the two-color cellular automata shown on previous pages. Note that in the sequence of rules shown here, those that change the white background are not included. The symmetry of all the patterns is a consequence of the basic structure of totalistic rules.