Note that the pattern in case (a) does eventually repeat, while the one in case (b) eventually shows a nested structure.

In case (c), the above connection of each node is rerouted so as to loop back to the node itself, while the below connection is left unchanged. And in case (d), the above connection is rerouted so as to loop back, while the below connection is rerouted to lead to the node reached by following the above connection.

But while one can emulate each step in the evolution of a mobile automaton or a Turing machine with a single step of cellular automaton evolution, this is no longer in general true for substitution systems. That this must ultimately be the case one can see from the fact that the total number of elements in a substitution system can be multiplied by a factor from one step to the next, while in a cellular automaton the size of a pattern can only ever increase by a fixed amount at each step.

The instruction being executed is indicated at each step by the position of the dot on the left, while the numbers in each of the two registers are indicated by the gray blocks on the right. … And in the first machine, the first register alternates between 0 and 1, while the second remains zero.

The angle of the beginning of the black sector in the clock indicates when the node was created, while the angle of its end represents the current step, so that older nodes have larger black sectors.

Order dependence [in symbolic systems] The operation expr /. lhs  rhs in Mathematica has the effect of scanning the functional representation of expr from left to right, and applying rules whenever possible while avoiding overlaps. (Standard evaluation in Mathematica is equivalent to expr //. rules and uses the same ordering, while Map uses a different order.)

It is for example known that (d) never reaches zero, while curve (c) reaches zero only for numbers of the form 4 r (8s + 7) .

For while at a large scale these may provide a reasonable description of average behavior in a network, it is almost inevitable that closer to the scale of individual connections they will have to be modified. … And while some aspects of this

image in a certain part of the area is, say, colored white, while others will be positive if it is colored black. … In each case the first picture shows the image on the retina, while the second picture shows which cells respond to it.

rules 0 and 128 all the cells become white, while in rule 255 all of them become black. … But while repetitive patterns are by a small margin the most common kind, about 14% of all the cellular automata shown yield more complicated kinds of patterns.