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[No text on this page] Evolution of one-dimensional slices through some of the two-dimensional cellular automata from the previous two pages [ 173 , 174 ]. Each picture shows the colors of cells that lie on the one-dimensional line that goes through the middle of each two-dimensional pattern.
[No text on this page] Minimal Boolean expression representations for the results of steps 1 through 5 in the evolution of three elementary cellular automata. … (For steps 1 through 6, the expressions involve 3, 7, 17, 41, 102 and 261 terms respectively.)
For assuming that around the particle there is some kind of uniformity in the causal network—and thus in the apparent structure of space—taking slices through the causal network at an appropriate angle will always make any particle appear to be at rest. … What will the events associated with the second particle look like if one takes slices through the causal network so that the first particle appears to be at rest? … With different slices through the causal network, the apparent size of this coat can change.
But after searching through perhaps 50,000 rules, one finally comes across a rule of the kind shown below—in which the compressed pattern exhibits very much the same kind of apparent randomness that we saw in cellular automata like rule 30. … But after searching through a few million rules, I finally found the example shown on the facing page .
It took searching through a few million mobile automata to find one with behavior as complex as what we see here.
These structures could later cause trouble, but looking at region (b) we see that in fact they just pass through other structures that they meet without any adverse effect. Region (c) shows what happens when the information corresponding to one element in a block passes through the kind of object produced in region (a). … Starting around the middle of the region, however, the behavior becomes quite different from region (a): while region (a) yields an object that allows information to pass through, region (g) yields one that stops all information, as shown in regions (h) and (i).
[No text on this page] One-dimensional slices through the evolution of various two-dimensional cellular automata.
Nevertheless, if one looks carefully at examples (a) through (h) each of them shows large regions of either repetitive or nested behavior. … But looking at 4-state 2-color Turing machines examples (i) through (l) again appear to exhibit roughly exponential growth. … And certainly if one allows no more than 4-state 2-color Turing machines I have been able to establish by explicitly searching all 4 billion or so possible rules that there is absolutely no way to speed up the computations in pictures (i) through (l).
From the universality of rule 110 we know that if one just starts enumerating cellular automata in a particular order, then after going through at most 110 rules, one will definitely see universality. And from other results earlier in this chapter it seems likely that in fact one would tend to see universality even somewhat earlier—after going through only perhaps just ten or twenty rules.
The maps shown can be thought of as being made by taking an infinitely dense limit of the array of pictures on the facing page , but keeping only what one sees in each picture by looking through a peephole at a particular position relative to the original stem.
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