But searching all 4 billion or so possible such systems with 2 × 2 blocks and up to four colors one finds not a single case in which a nested pattern is forced to occur.

In practical trading, partly as an outgrowth of theories of business cycles, there had emerged all sorts of elaborate so-called technical analysis in which patterns of price movements were supposed—often on the basis of almost mystical theories—to be indicators of future behavior.

Following the publication of Charles Darwin 's Origin of Species in 1859 many scientists began to argue that natural selection could explain all the basic phenomena of biology, and although some religious groups maintained strong resistance, it was widely assumed by the mid-1900s that no other explanation was needed.

In general, however, a 1D cellular automaton rule can be given as a set of explicit replacements for all possible blocks of cells in each neighborhood (see page 60 ).

But even now, unlike essentially all other branches of science, mainstream mathematics continues to be entirely dominated by theoretical rather than experimental methods.

If f has an identity element, so that f[1, i]  i for all i , and has inverses, so that f[i, j]  1 for some j , then the system is a group.

But as discussed in the main text I strongly suspect that in fact there are no such global constraints, and instead almost all reasonable values of parameters from the simple model used do actually occur in real molluscs.

But for the most part I concentrated on characterizing behavior obtained from all possible initial conditions. … I do not know exactly what made me start looking more carefully at simple initial conditions, though I believe that I first systematically generated high-resolution pictures of all the k = 2 , r = 1 cellular automata as an exercise for an early laserprinter—probably at the beginning of 1984. … But cellular automata—and especially 1D ones—make the phenomena particularly clear, which is why even after investigating all sorts of other systems 1D cellular automata are still the most common examples that I use in this book.

The types of machines discussed so far all have the feature that they have to be physically rearranged or rewired in order to perform different calculations. … All the systems mentioned so far had the feature that they performed operations in what was essentially a fixed sequence.

And from the 1920s to the 1950s Albert Einstein increasingly considered the possibility that there might be a unified field theory in which all matter would somehow be associated with the geometry of space. … So while it remains impossible to work out all the consequences of string theories, it is conceivable that among the representations of such theories there might be ones in which matter can be viewed as just being associated with features of space.