Among the 4,294,967,296 possible next-neighbor rules, only 428 exhibit the kind of conservation property shown here.

For not only can it emulate any cellular automaton: it can also emulate any of a wide range of other systems, including practical computers.

The emulation here is, however, only for a single step.

The width of the pattern in these cases after t steps grows roughly like √ 2t . Compressed versions of the evolution of mobile automata (g) and (h) above, obtained by showing only those steps at which the active cell is further to the left or right than it has ever been before.

In every picture both x and y run from 1 to 127. d[n] stands for IntegerDigits[n, 2, 7] . (h) is equivalent to digit sequences of powers of 3 in base 2 (see page 120 ). … The only pattern that is known to be obtainable by evolving down the page according to a simple local rule is (j), which corresponds to the rule 60 elementary cellular automaton.

The successive blocks of results in each case show forms of the operator allowed with 2, 3 and 4 possible elements. Note that with 3 and 4 elements, only forms inequivalent under interchange of element labels are shown. … In each case the operator forms shown correspond to possible semigroups, commutative groups, systems of logic (Boolean algebras), etc. with 2, 3 and 4 possible elements.

Nand and Nor are the only primitive functions that work on their own.

Note that of the 8 cases in the basic rule for rule 110, only one differs from rule 102—which is a simple additive rule obtained by reflecting rule 60.

K 0 decay was for a long time the only example of time reversal violation that had explicitly been seen, although recently examples in B particle decays have probably also been seen. … The basic formalism used for particle physics assumes not only reversibility, but also so-called CPT invariance. … Starting with a pure beam of K 0 particles, it turns out that quantum mechanical mixing processes lead after about 10 -8 seconds to a certain mixture of K 0 particles—the antiparticles of the K 0.

But it is certainly not the case that all two-dimensional cellular automata produce only simple overall shapes. … And so it seems that there can be great complexity not only in the detailed arrangement of black and white cells in a two-dimensional cellular automaton pattern, but also in the overall shape of the pattern.