Of all k k 2r + 1 rules with k colors and range r it turns out that there are always exactly k 2r + 1 additive ones—each obtained by taking the cells in the neighborhood and adding them modulo k with weights between 0 and k - 1 . … Note that each step in the evolution of any additive cellular automaton can be computed as Mod[ListCorrelate[w, list, Ceiling[Length[w]/2]], k] (See page 1087 for a discussion of partial additivity.)

The Central Limit Theorem leads to a self-similarity property for the Gaussian distribution: if one takes n numbers that follow Gaussian distributions, then their average should also follow a Gaussian distribution, though with a standard deviation that is 1/ √ n times smaller.

An example that depends on three neighbors on each side was discovered by Peter Gacs , Georgii Kurdyumov and Leonid Levin in 1978, following work on how reliable electronic circuits can be built from unreliable components by Andrei Toom : {a1_, a2_, a3_, a4_, a5_, a6_, a7_}  If[If[a4  1, a1 + a3 + a4, a4 + a5 + a7] ≥ 2, 1, 0] The 4-color rule shown in the text is probably the clearest example available in one dimension.

However, the ordering defined by GrayCode from page 901 does do this for one particular sequence of single square changes.

General constraints on growth Given a system made from material with certain overall properties, one can ask what distributions of growth are consistent with those properties, and what kinds of shapes can be produced.

Initial conditions [for the universe] To find the behavior of the universe one potentially needs to know not only its rule but also its initial conditions.

A procedure analogous to the one on page 350 was introduced by Charles Bennett in 1971 for 3D spheres (relevant for binary alloys). … Note that with the procedure used, each new circle added must immediately touch two existing ones, though subsequently it may get touched by varying numbers of other circles. … The picture below shows what happens if one repeatedly inserts circles to form a so-called Apollonian packing derived from the problem studied by Apollonius of finding a circle that touches three others.

But to describe things like fields one must allow particles to be created and destroyed. In the mid-1920s there was already discussion of how to set up a formalism for this, with an underlying idea again being to think in terms of virtual oscillators—but now one for each possible state of each possible one of any number of particles. … (A typical goal is to find variables in which one can carry out what is known as canonical quantization: essentially applying the same straightforward transformation of equations that happens to work in ordinary elementary quantum mechanics.)

(Typically one needs to generalize formulas that are initially set up with integer numbers of terms; examples include taking Power[x, y] to be Exp[Log[x] y] and x! … And if one modifies the usual hypergeometric equation y''[x]  f[y[x], y'[x]] by making f nonlinear then solutions typically become hard to find, and vary greatly in character with the form of f .

To emulate cellular automaton evolution one starts by encoding a list of cell values by the single combinator p[num[Length[list]]][ Fold[p[Nest[s, k, #2]][#1] &, p[k][k], list]] //. crules where num[n_] := Nest[inc, s[k], n] inc = s[s[k[s]][k]] One can recover the original list by using Extract[expr, Map[Reverse[IntegerDigits[#, 2]] &, 3 + 59(16^Range[Depth[expr[s[k]][s][k] //. crules] - 1, 1, -1] - 1)/ 15)]]/. {k  0, s[k]  1} In terms of the combinator f a single complete step of cellular automaton evolution can be represented by w = cr[p[inc[inc[x[s[k]]]]][ inc[x[s[k]]][cr[p[y[s[k]][k]][p[y[s[k]][s[k]]][y[k]]], {y}]][p[x[s[k]][cr[p[p[f[y[k][k][k][s[k]]][ y[k][k][s[k]]][y[k][s[k]]]][y[s[k]]]][y[k][k]], {y}]][ p[p[k][k]][p[k][x[k]]]][s[k]]][p[k][p[k][k]]]][k]], {x}] cr[expr_, vars_] := ToC[expr //. crules, vars] where there is padding with 0 on either side. With this setup t steps of evolution are given simply by Nest[w, init, t] .