Complex powers [of numbers] The pictures below show successive powers of complex numbers z with digits extracted according to (2 d[Re[#], w] + d[Im[#], w]) & [z t ] d[x_, w_] := If[x < 0, 1 - d[-x, w], IntegerDigits[x, 2, w]] Non-trivial cases of complex number multiplication never correspond to local cellular automaton operations.

If the underlying cellular automaton rule exhibits an invariance—say under reflection in space or permutation of colors—this will also often lead to the presence of identical pieces in the final network, corresponding to cosets of the symmetry transformation.

The two cellular automata below both have all white and all black as invariant states. … The two-dimensional patterns that arise from the constraints at the end of Chapter 5 all turn out to correspond to invariant states of various two-dimensional cellular automata. And so for example the pattern of page 211 is found to be the unique invariant state for 572,522 of the 4,294,967,296 possible five-neighbor cellular automaton rules.

Nonlinear feedback shift registers Linear feedback shift registers of the kind discussed on page 974 can be generalized to allow any function f (note the slight analogy with cyclic tag systems): NLFSRStep[f_, taps_, list_] := Append[Rest[list], f[list 〚 taps 〛 ]] With the choice f=IntegerDigits[s, 2, 8] 〚 8 - # . {4, 2, 1} 〛 & and taps = {1, 2, 3} this is essentially a rule s elementary cellular automaton. With a list of length n , Nest[NLFSRStep[f, taps, #] &, list, n] gives one step in the evolution of the cellular automaton in a register of width n , with a certain kind of spiral boundary condition. … Tap positions {1, 2, 3, 4} were among those studied, but nothing like the pictures below were apparently ever explicitly generated—and nearly three decades passed before I noticed the remarkable behavior of the rule 30 cellular automaton.

There are however a few cellular automata in which class 3 behavior is obtained with random initial conditions, but in which no significant randomness is ever produced with simple initial conditions. … This particular cellular automaton rule has the special property of additivity which implies that with any initial conditions the patterns that it produces can be obtained as simple superpositions of the first pattern shown above.

For if one was to work out the value of a power m t by explicitly performing t multiplications, this would be very similar to explicitly following t steps of cellular automaton evolution. … So what about other patterns produced by cellular automata and similar systems?

Note (d) for Cellular Automata…History [of 2D cellular automata] As indicated on pages 876 – 878 , 2D cellular automata were historically studied more extensively than 1D ones—though rarely with simple initial conditions. … But until after my work in the 1980s simulators for more general 2D cellular automata were rare. … And as mentioned on page 1077 , going back to the 1950s some image processing systems have been based on particular families of 2D cellular automaton rules.

Cellular automata [as randomness generators] From the discussion here it should not be thought that in general there is necessarily anything better about generating randomness with cellular automata than with systems based on numbers. … In addition, one should recognize that while the complete evolution of the cellular automaton may effectively generate perfect randomness, there may be deviations from randomness introduced when one constructs a practical random number generator with a limited number of cells.

Note (d) for Cellular Automata…General rules [for multidimensional cellular automata] One can specify the neighborhood for any rule in any dimension by giving a list of the offsets for the cells used to update a given cell. … If a cellular automaton rule takes the new color of a cell with neighborhood configuration IntegerDigits[i, k, Length[os]] to be u 〚 i + 1 〛 , then one can define its rule number to be FromDigits[Reverse[u], k] . A single step in evolution of a general cellular automaton with state a and rule number num is then given by Map[IntegerDigits[num, k, k^Length[os]] 〚 -1 - # 〛 &, Apply[Plus, MapIndexed[k^(Length[os] - First[#2]) RotateLeft[a, #1] &, os]], {-1}] or equivalently by Map[IntegerDigits[num, k, k^Length[os]] 〚 -# - 1 〛 &, ListCorrelate[Fold[ReplacePart[k #1, 1, #2 + r + 1] &, Array[0 &, Table[2r + 1, {d}]], os], a, r + 1], {d}]

Partial Differential Equations By introducing continuous cellular automata with a continuous range of gray levels, we have successfully removed some of the discreteness that exists in ordinary cellular automata. But there is nevertheless much discreteness that remains: for a continuous cellular automaton is still made up of discrete cells that are updated in discrete time steps.