Density in rule 90 From the superposition principle above and the number of black cells at step t in a pattern starting from a single black cell (see page 870 ) one can compute the density after t steps in the evolution of rule 90 with initial conditions of density p to be (see also page 602 ) 1/2 (1 - (1 - 2 p)^(2^DigitCount[t,2,1]))

If nodes can have more than three connections, then they will often be able to evolve to have any number of connections—in which case one must give what is in effect an infinite set of rules to specify what to do for each number of connections.

It was at first hoped that the problems could be NP-complete ones, which are universal in the sense that their solution can be used to provide a solution to any problem in the class NP (see page 1086 ). … But the greater simplicity of rule 30 might make one already have almost as much confidence in the difficulty of solving this problem as of factoring integers.

Cellular automaton axioms The first 4 axioms are general to one-dimensional cellular automata. … One can establish that the statement at the bottom on the right cannot be proved either true or false from the axioms by showing that it is true for some initial conditions and false for others.

But given {a 1 , 0, a 2 , 0, a 3 , 0} the value after one step is {Mod[a 1 + a 2 , 2], 0, Mod[a 2 + a 3 , 2], 0} and after two steps is again {Mod[a 1 + a 3 , 2], 0} . … And it follows that in this case the pattern generated after a certain number of steps from a single non-white cell will always be the same as one gets by going k times that number of steps and then keeping only every k th row and column.

In a plant as large as a typical tree—particularly one that grows slowly—different conditions associated with the growth of different branches may however destroy some of the regularity of branching. … Monocotyledons—of which palms and grasses are two examples—typically have only one primary site of growth, and thus do not exhibit repeated branching.

One study was done in 1969 by Conrad Waddington and Russell Cowe in which patterns on one particular kind of shell were reproduced by a specific computer simulation based on the idea of diverging waves of pigment.

More steps in the evolution on the previous page , with opening brackets represented by dark squares and closing brackets by light ones.

Each program in the sequence differs from the previous one by a single mutation, made completely at random.

Just as for substitution systems on strings, one can find causal networks that represent the causal connections between different updating events on networks.