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.

The top pictures show step 30,000, while those on the bottom right show step 200.

So as an example of all this consider cellular automata that achieve the purpose of doubling the width of the pattern given in their input. … But while this cellular automaton seems to have little extraneous going on, it operates in a slow and sequential way, and its underlying If the purpose is to generate a uniformly expanding pattern it seems more plausible that the top cellular automaton should have been the one created for this purpose.

Then it is cut in two, and the two halves are stacked on top of each other.

At the top in each case is shown the explicit array of outcomes for each of the 32 possible initial configurations of cells.

(The function s[d] has a maximum around d = 5.26 , then decreases rapidly with d .) … Such circles are no longer flat, but instead are like caps on the sphere—with a circle of radius r containing all points that are geodesic (great circle) distance less than r from its center.

The outer walls of pollen grains are often covered with a certain density of tiny columns that can form spikes, or can have plates on top that can form cross-linkages and can join together to appear as patches.

But explicit construction, based on correspondence with one-dimensional cellular automata, leads to the example shown at the top of the facing page : a system with 56 allowed templates in which the only pattern satisfying the constraint is a complex and largely random one, derived from the rule 30 cellular automaton.

The pictures at the top of the next page show what happens in this case.

Here again it turns out to be fairly easy to emulate cellular automata—as the pictures at the top of the facing page demonstrate.