In any 1D k = 2 , r = 1 cellular automaton, it follows from the basic structure of the rule that one can tell what the difference in values of a particular cell on two successive steps will be just by looking at the cell and its immediate neighbor on each side. But if the number of black cells is conserved, then one can compute this difference instead by defining a suitable flux, and subtracting its values on the left and right of the cell.

2D run-length encoding A simple way to generalize run-length encoding to two dimensions is to scan data one row after another, always finding the largest rectangle of uniform color that starts at each particular point.

[Computational complexity of] finding outcomes If one sets up a function to compute the outcome after t steps of evolution from some fixed initial condition—say a single black cell in a cellular automaton—then the input to this function need contain only Log[2, t] digits.

But as soon as one introduces actual axioms that constrain the operators this is no longer true—and in general it can be undecidable whether or not a particular equivalence holds.

Recognition of art One bizarre possibility is that forms like those from rule 30 could have been created as art long ago but not be recognized now.

In case (b), with rows chosen to be 2 j elements in length, the leftmost column will always be identical to the beginning of the sequence, and in addition every interior element will be black exactly when the cell at the top of its column has the same color as the one at the beginning of its row.

But at some level one always gives discrete symbolic descriptions of the logical structure of programs.

Implementation [of hexagonal cellular automata] One can treat hexagonal lattices as distorted square lattices, updated according to CAStep[rule_List, a_] := Map[rule 〚 14 - # 〛 &, a + 2 ListConvolve[{{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}, a, 2], {2}] where rule = IntegerDigits[code, 2, 14] .

These can be modelled by having a cellular automaton in which one starts from several separated seeds.

Adaptive value of complexity One might think that the reason complexity is not more widespread in biology is that somehow it is too sensitive to perturbations.