Reproducing quantum phenomena Given molecular dynamics it is much easier to see how to reproduce fluid mechanics than rigid-body mechanics—since to get rigid bodies with only a few degrees of freedom requires taking all sorts of limits of correlations between underlying molecules.

By the early 1800s, such ideas had led to the field of natural theology, and William Paley gave the much quoted argument that if it took a sophisticated human watchmaker to construct a watch, then the only plausible explanation for the vastly greater complexity of biological systems was that they must have been created by a supernatural being. … In fact, however, just how complexity arises was never really resolved, and in the end I believe that it is only with the ideas of this book that this can successfully be done.

Only if these rules have the confluence property will the results always be unique, and independent of the order of rule application. … Showing only the arguments to f , the pictures below illustrate how the flat functions Xor and And are confluent, while the non-flat function Implies is not.

(GPS P-code apparently uses much longer LFSR sequences and repeats only every 267 days.

But the comparative weakness of natural selection (see page 391 ) has meant that only a limited set of such forms have actually been explored.

Numbers of possible [2D cellular automaton] rules The table below gives the total number of 2D rules of various types with two possible colors for each cell. … Totalistic rules depend only on the total number of black cells in a neighborhood; outer totalistic rules (as in the previous note) also depend on the color of the center cell. … In such a rule, given a list of how many neighbors around a given cell (out of s possible) make the cell turn black the outer totalistic code for the rule can be obtained from Apply[Plus, 2^Join[2 list, 2 Range[s + 1] - 1]]

Note (c) for Systems of Limited Size and Class 2 Behavior…The smallest such t is given by MultiplicativeOrder[k, n] , which always divides EulerPhi[n] (see page 1093 ), and has a value between Log[k, n] and n - 1 , with the upper limit being attained only if n is prime. … When GCD[k, n]  1 the dot can never visit position 0. But if n  k s , the dot reaches 0 after s steps, and then stays there.

With material that is completely rigid growth can occur only at boundaries. … With material where parts can locally expand, but cannot change their shape, page 1007 showed that a 2D surface will remain flat if the growth rate is a harmonic function.

If one drops the requirement of cells going into a special state, then even the 2-color elementary rule 60 shown on the left can be viewed as solving the problem—but only for widths that are powers of 2.

One can also consider conservation of a vector quantity such as momentum which has not only a magnitude but also a direction. Direction makes little sense in 1D, but is meaningful in 2D. The 2D cellular automaton used as a model of an idealized gas on page 446 provides an example of a system that can be viewed as conserving a vector quantity.