Class 3 tends to become more common as the number of elements in the rule increases because as soon as any of these elements yield class 3 behavior, that behavior dominates the system.

But for the purpose of nearest-neighbor cellular automaton rules, what matters is not detailed geometry, but merely what cells are adjacent to a given cell. … (Outer totalistic codes specify rules; the first rule makes a particular cell black when any of its five neighbors are black and has code 4094. … This tiling has two different shapes of tile, but here both are treated the same by the cellular automaton rule, which is given by an outer totalistic code number.

Probabilistic cellular automata As an alternative to having continuous values at each cell, one can consider ordinary cellular automata with discrete values, but introduce probabilities for, say, two different rules to be applied at each cell.

Rule structure [for network systems] For depth 1, the possible results from NeighborNumbers are {1} and {2} .

This is a k = 8 2D cellular automaton in which toppling of sand above a critical slope is captured by updating an array of relative sand heights s according to the rule SandStep[s_]:= s + ListConvolve[ {{0, 1, 0}, {1, -4, 1}, {0, 1, 0}}, UnitStep[s - 4], 2, 0] Starting from any initial condition, the rule eventually yields a fixed configuration with all values less than 4, as in the picture below. … In 1D, the update rule is simply SandStep[s_] := s + ListConvolve[{1, -2, 1}, UnitStep[s - 2], 2, 0] In this case the evolution obtained if one repeatedly adds to the center cell (as in the first picture below) is always quite simple. But as the pictures below illustrate, evolution from typical initial conditions yields behavior that often looks a little like rule 184.

. • 1200s: Fibonacci sequences, Pascal's triangle and other rule-based numerical constructions are studied, but are found to show only simple behavior. • 1500s: Leonardo da Vinci experiments with rules corresponding to simple geometrical constraints (see page 875 ), but finds only simple forms satisfying these constraints. • 1700s: Leonhard Euler and others compute continued fraction representations for numbers with simple formulas (see pages 143 and 915 ), noting regularity in some cases, but making no comment in other cases. • 1700s and 1800s: The digits of π and other transcendental numbers are seen to exhibit apparent randomness (see page 136 ), but the idea of thinking about this randomness as coming from the process of calculation does not arise. • 1800s: The distribution of primes is studied extensively—but mostly its regularities, rather than its irregularities, are considered. … (See page 999 .) • 1956-1959: Solomon Golomb simulates nonlinear feedback shift registers—some with rules close to rule 30—but studies mainly their repetition periods not their detailed complex behavior. (See page 1088 .) • 1960, 1967: Stanislaw Ulam and collaborators simulate systems close to 2D cellular automata, and note the appearance of complicated patterns (see above ). • 1961: Edward Fredkin simulates the 2D analog of rule 90 and notes features that amount to nesting (see above ). • Early 1960s: Students at MIT try running many small computer programs, and in some cases visualizing their output.

For basic logic, examples of this were studied in the mid-1900s, and with the transformations thought of as rules of inference they were sometimes known as "axiomless formulations".

Localized structures [in Turing machines] Even when the overall behavior of a Turing machine is complicated, it is possible for simple localized structures to exist, much as in cellular automata such as rule 110.

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).

Meaning of the universe If the whole history of our universe can be obtained by following definite simple rules, then at some level this history has the same kind of character as a construct such as the digit sequence of π .