The networks are laid out in analogy to the space networks on page 479 , with nodes being placed on successive rows if they take progressively more connections to reach from the top node.

Mobile Automata One of the basic features of a cellular automaton is that the colors of all the cells it contains are updated in parallel at every step in its evolution. … Much as for cellular automata, one can enumerate all possible rules of this kind; it turns out that there are 65,536 of them. The pictures at the top of the next page show typical behavior obtained with such rules.

As the top row of pictures in the second image below demonstrate, the details depend on the geometry and relative growth rates of new elements and of the original stem. But in all cases very characteristic patterns are produced.

In each close-up the pattern grows from top to bottom, just like in a one-dimensional cellular automaton. … The patterns are all various shades of brown on roughly white backgrounds.

[No text on this page] Maps of where in the space of parameters for the substitution systems on the facing page the patterns obtained overlap the region indicated in the icon at the top left of each picture.

But just as in so many other situations that we have seen in this book, what we have seen is that in fact a very simple rule is all that is in the end needed. … In the top row the disks are always flat, forcing the cells of material to vary in size and shape. In the bottom row, the disks form shapes in three dimensions in which all cells are the same size and shape.

[No text on this page] Results obtained by deducing optimal orderings of basic forms from collections of images of cellular automata (top) and letters (bottom).

so long as the structure of the network is kept the same, it is fairly easy even in this case to deduce from a given set of data what probabilities in the network provide the best model for the data—for essentially all one need do is to follow the path corresponding to the data, and see with what frequency each connection from each node ends up being used. … Because the rule is probabilistic a different detailed pattern of evolution will in general be obtained each time the cellular automaton is run—as in the top row of pictures above.

The top left picture shows a solution to the wave equation. The top right picture shows a solution to an equation obtained from the wave equation by transforming the time variable according to t->1-1/t .

The initial condition for the system is specified by the position of the incoming light ray in the gray region at the top of each picture.