The pictures include all 14 templates that involve nodes out to distance at most two for which complete networks can be formed.

All their elements, for example, are always arranged in a rigid array, and are always updated in parallel at each step. … And the remarkable conclusion is that in the end none of these features actually matter much at all.

The examples of complexity that I have shown so far in this book are almost all completely new. … But in fact with considerable effort it has been proved that all of them are in a sense more random—and eventually cross the axis an infinite number of times, and indeed go any distance up or down.

Cellular Automata The cellular automata that we have discussed so far in this book are all purely one-dimensional, so that at each step, they involve only a single line of cells. But one can also consider two-dimensional cellular automata that involve a whole grid of cells, with the color of each cell being updated according to a rule that depends on its neighbors in all four directions on the grid, as in the picture below.

In the two cases shown as blank rectangles on the upper right, there are no patterns at all that satisfy the constraints. … But ultimately, in every case where some pattern can work, a simple repetitive pattern is all that is needed.

A more practical alternative is to build up patterns iteratively, starting with a small region, and then adding new cells in essentially all possible ways, at each stage backtracking if the constraint for the system does not end up being satisfied. … And what if there is no pattern at all that can satisfy a particular constraint?

The right-hand neighbor of the rightmost cell in any particular block is the leftmost cell in the next block, but since all the blocks are identical, this cell always has the same color as the leftmost cell in the block itself. … It turns out that no block of any size gives a period of exactly two steps, but blocks can be found for all larger periods at least up to 15 steps.

Indeed, all it does is to cause digits which make an arbitrarily small contribution to the size of numbers in the initial conditions eventually to have a significant effect. … And as discussed on page 152 , what this idealization suggests is that all numbers which are sufficiently close in size should somehow be equally common.

But the special feature of the cellular automata shown on the facing page is that they have two very different stable states—either all white or all black—and when one changes the initial density a discrete transition occurs between these two states.

When all circles are the same size, this procedure yields a simple repetitive pattern. … One can look at all sorts of other physical systems, but so far as I can tell the story is always more or less the same: whenever there is behavior of significant complexity its most plausible explanation tends to be some explicit process of evolution, not the implicit satisfaction of constraints.