Frequencies of [cellular automaton] classes The pie charts below show results for 1D totalistic cellular automata with k colors and range r .

Mathematical interpretation of cellular automata In the context of pure mathematics, the state space of a 1D cellular automaton with an infinite number of cells can be viewed as a Cantor set. The cellular automaton rule then corresponds to a continuous mapping of this Cantor set to itself (continuity follows from the locality of the rule).

Additive [cellular automaton] rules Of the 256 elementary cellular automata 8 are additive: {0, 60, 90, 102, 150, 170, 204, 240} . … Note that each step in the evolution of any additive cellular automaton can be computed as Mod[ListCorrelate[w, list, Ceiling[Length[w]/2]], k] (See page 1087 for a discussion of partial additivity.)

The Notion of Computation Computation as a Framework In earlier parts of this book we saw many examples of the kinds of behavior that can be produced by cellular automata and other systems with simple underlying rules. … Throughout this book I have referred to systems such as cellular automata as simple computer programs. … In a typical case, the initial conditions for a system like a cellular automaton can be viewed as corresponding to the input to a computation, while the state of the system after some number of steps corresponds to the output.

Cellular automaton [Nand] formulas For 1 step, the elementary cellular automaton rules are exactly the 256 n = 3 Boolean functions.

More complicated rules [and reducibility] The standard rule for a cellular automaton specifies how every possible block of cells of a certain size should be updated at every step. … Note that dealing with blocks of different sizes requires going beyond an ordinary cellular automaton rule.

Cellular Automata…[No text on this page] A cellular automaton that yields a pattern whose shape closely approximates a circle.

The Threshold of Universality in Cellular Automata…At the end of the last section I mentioned rule 54 as another elementary cellular automaton besides rule 110 that might be class 4.

Note (a) for Cellular Automata…Limiting shapes [in 2D cellular automata] When growth occurs at the maximum rate the outer boundaries of a cellular automaton pattern reflect the neighborhood involved in its underlying rule (in rough analogy to the Wulff construction for shapes of crystals).

Note (e) for More Cellular Automata…Mod 3 [cellular automaton] rule Code 420 is an example of an additive rule, and yields a pattern corresponding to Pascal's triangle modulo 3, as discussed on page 870 .