For equations of the form ∂ tt u[t, x]  ∂ xx u[t, x] + f[u[t, x]] one can set up a simple finite difference method by taking f in the form of pure function and creating from it a kernel with space step dx and time step dt : PDEKernel[f_, {dx_, dt_}] := Compile[{a,b,c,d}, Evaluate[(2 b - d) + ((a + c - 2 b)/dx 2 + f[b]) dt 2 ]] Iteration for n steps is then performed by PDEEvolveList[ker_, {u0_, u1_}, n_] := Map[First, NestList[PDEStep[ker, #]&, {u0, u1}, n]] PDEStep[ker_, {u1_, u2_}] := {u2, Apply[ker, Transpose[ {RotateLeft[u2], u2, RotateRight[u2], u1}], {1}]} With this approach an approximation to the top example on page 165 can be obtained from PDEEvolveList[PDEKernel[ (1 - # 2 )(1 + #)&, {.1, .05}], Transpose[ Table[{1, 1} N[Exp[-x 2 ]], {x, -20, 20, .1}]], 400] For both this example and the middle one the results converge rapidly as dx decreases.

[Networks generated by] random replacements As indicated in the note above, applying the second rule (T1, shown as (b) on page 511 ) at an appropriate sequence of positions can transform one planar network into any other with the same number of nodes. … Each time it is applied, the rule adds two edges to one face, and removes them from another.

Games between programs One can set up a game between two programs generating single bits of output by for example taking the input at each step to be the concatenation of the historical sequences of outputs from the two programs.

Note that 2n has the same digits as n , but shifted one position to the left.

(With just one block the behavior is always repetitive.)

Runs of digits [in numbers] One can consider any base 2 digit sequence as consisting of successive runs of 0's and 1's, constructed from the list of run lengths by Fold[Join[#1, Table[1 - Last[#1], {#2}]] &, {0}, list] This representation is related to so-called surreal numbers (though with the first few digits different).

One such principle implies that atoms in molecules will tend to arrange themselves so as to minimize their energy.

Rate equations In standard chemical kinetics one assumes that molecules are uniformly distributed in space, so that the rates for particular reactions are proportional to the products of the densities of the molecules that react in them.

The same is true even if one allows a small fraction of squares to violate the constraints.

With all cells 0 on one step, and a block of nonzero cells on the next step, the periods are for example: {1} : 21 ; {1, 1} : 3n - 8 ; {1, 0, 1} : 666 ; {1, 1, 1} : 3n - 8 ; {1, 0, 0, 1} : irregular ( < 24n ; peaks at 6j + 1 ); {1, 0, 0, 1, 0, 1} : irregular ( ≲ 2 n ; 857727 for n= 2 6 ; 13705406 for n = 100 ).