One approach—which turns out to be similar to what is used in practice in most current high-performance general-purpose compression systems—is to set up an encoding in which any particular sequence of elements above some length is given explicitly only once, and all subsequent occurrences of the same sequence are specified by pointers back to the first one. … In the examples shown, pointers are used only for sequences of length at least 6.

This is a consequence of the fact that the cellular automaton rule allows only certain blocks to appear in the pattern, as illustrated in the picture below. … Cellular automaton rule 30, and the 3×2 blocks which appear in large patterns generated by it. There are a total of 2 6 =64 possible 3×2 blocks of black and white cells; the fact that only 24 of them appear in patterns generated by rule 30 is what makes it possible for two-dimensional block-based encoding to compress such patterns.

Among 4-state 2-color Turing machines the same kind of complex behavior is also seen—as discussed on page 81 —but now it occurs only in perhaps one out of 200,000 cases. … There are a total of 2,985,984 of these. … The compressed picture above is made by taking the first 100,000 steps, and keeping only those at which the head is further to the left than ever before.

Or does the notion of computation somehow apply only to systems with abstract elements like, say, the black and white cells in a cellular automaton? … And indeed in the end the only unfamiliar aspect of this is that the rules such processes follow are defined not by some computer program that we as humans construct but rather by the basic laws of nature.

But whereas cellular automata always evolve only in discrete steps, differential equations instead go through a continuous process of evolution in which time appears just as a parameter. And by making simple algebraic changes to the way that time enters a differential equation one can often arrange, as in the pictures below, that processes that would normally take an infinite time will actually always occur over only a finite time.

With the particular rule shown, the behavior always eventually stabilizes—though sometimes only after an astronomically long time.

In the second set of pictures, the rule specifies that a cell should become black only when exactly one of its six neighbors was black on the step before.

In the bottom pictures, the rule specifies that a cell should become black only when exactly two of its 26 neighbors were black on the step before.

In cases where the underlying rules have only rather simple behavior—as with rules 90 and 184—it turns out that it is never possible to emulate more than a Illustrations of how rule 30 can be set up to emulate a single step in the evolution of all elementary cellular automata.

For light level—as with color constancy—this is presumably achieved by responding only to differences between levels at different positions. Probably the same effect contributes to scale invariance by emphasizing only edges and corners. … With good lighting and good eyesight the textures in the picture can still be distinguished at a distance of 5 feet, where each square covers only one cell.