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Note (b) for How Do Simple Programs Behave?

Nesting occurs in all cellular automata with additive rules (see page 955 ).

Note (b) for Continuous Cellular Automata

If k is irrational then equidistribution of Mod[Binomial[t, x], k] implies that all possible values eventually appear; the corresponding patterns seem fairly irregular, as shown below.

Note (b) for Substitution Systems and Fractals

Connection [of geometric substitution systems] with digit sequences Patterns after t steps can be viewed as containing all t -digit integers in an appropriate complex base.

Note (e) for Network Systems

Cellular automata on networks The cellular automata that we have considered so far all have cells arranged in regular arrays.

Note (a) for Growth of Plants and Animals

In plants, as we have discussed, essentially all cells have rigid cellulose walls.

Testing all 11,019,960,576 possible programs of length eight revealed just this and 125 similar cases of complex behavior.

An Outline of Basic Ideas

If theoretical science is to be possible at all, then at some level the systems it studies must follow definite rules.

How Do Simple Programs Behave?

For all one need do is just set up a sequence of possible simple programs, and then run them and see how they behave.

Randomness in Class 3 Systems

And is it perhaps all somehow just a reflection of randomness that was inserted in the initial conditions?

The Growth of Crystals

To a good approximation, all the molecules in a snowflake ultimately lie on a simple hexagonal grid.