If the cell has 1, 2 or 4 black neighbors, then it stays the same color as before, and if it has 5 or more black neighbors, then it becomes white on the next step.

Correspondence systems [as constraints] For a discussion of a class of 1D systems based on constraints see page 757 .

In the first two pictures below, bands with spacing 1/2 Csc[ θ /2] are visible wherever lines cross. … The patterns are exactly repetitive only when Tan[ θ ]  u/v , where u and v are elements of a primitive Pythagorean triple (so that u , v and Sqrt[u 2 + v 2 ] are all integers, and GCD[u, v]  1 ). … The second row of pictures illustrates what happens if points closer than distance 1/ √ 2 are joined.

But for multiway systems where each rule p  q is accompanied by its reverse q  p , and such pairs are represented say by "AAB" ↔ "BBAA" , an equivalent operator system can immediately be obtained either from Apply[Equal, Map[Fold[#2[#1] &, x, Characters[#]] &, rules, {2}], {1}] or from (compare page 1172 ) Append[Apply[Equal, Map[(Fold[f, First[#], Rest[#]] &)[Characters[#]] &, rules, {2}], {1}], f[f[a, b], c]  f[a, f[b, c]]] where now objects like "A" and "B" are treated as constants—essentially functions with zero arguments.

For large n this number is on average of order Log[n] + 2 EulerGamma - 1 . (b) (Aliquot sums) The quantity that is plotted is DivisorSigma[1, n] - 2n , equal to Apply[Plus, Divisors[n]] - 2n . … For d = 2 , this approaches π n for large n , with an error of order n c , where 1/4 < c ≤ 0.315 .

In a sense the fundamental reason for this—as we discussed on page 252 —is that such class 1 and class 2 cellular automata never allow any transmission of information except over limited distances. … These cellular automata are necessarily all class 1 or class 2 systems.

From the discussion of page 1024 one can reproduce the 1D diffusion equation with a continuous block cellular automaton in which the new value of each block is given by {{1 - ξ , ξ }, { ξ , 1 - ξ }} . {a 1 , a 2 } . … {a 1 , a 2 } .

One can have a rule be applied only once using Module[{i = 1}, expr /. lhs  rhs /; i++  1] Many symbolic systems (including the one on page 103 ) have the so-called Church–Rosser property (see page 1036 ) which implies that if a fixed point is reached in the evolution of the system, this fixed point will be the same regardless of the order in which rules are applied.

Cyclic tag systems which allow any value for each element can be obtained by adding the rule CTStep[{{r_, s___}, {n_, a___}}] := {{s, r}, Flatten[{a, Table[r, {n}]}]} The leading elements in this case can be obtained using CTListStep[{rules_, list_}] := {RotateLeft[rules, Length[list]], With[{n = Length[rules]}, Flatten[Apply[Table[#1, {#2}] &, Map[Transpose[ {rules, #}] &, Partition[list, n, n, 1, 0]], {2}]]]}

(For the cellular automaton on page 339 the simple condition for equilibrium is p  p 2 (3 - 2p) , which correctly implies that 0, 1/2 and 1 are possible equilibrium densities.)