To update the color of the cell represented by a particular block, what the universal cellular automaton must then do is to determine which of the 8 cases applies to that cell. And it does this by successively eliminating cases that do not apply, until eventually only one case remains.

The rules for updating such networks turn out to be somewhat more difficult to apply than those for the network systems discussed here.

Iterated bitwise operations The pictures below show digit sequences generated by repeatedly applying combinations of bitwise and arithmetic operations.

Implementing boundary conditions [in cellular automata] In the bitwise representation discussed on page 865 , 0's outside of a width n can be implemented by applying BitAnd[a, 2 n -1] at each step.

And if it is not, then results obtained by applying the rule can involve meaningless quantities such as f[0] , f[–1] and f[–2] .

Related [texture perception] models Rather than requiring particular templates to be matched, one can consider applying arbitrary cellular automaton rules.

(b) (Aliquot sums) The quantity that is plotted is DivisorSigma[1, n] - 2n , equal to Apply[Plus, Divisors[n]] - 2n . … The number of ways of expressing an integer n as the sum of two such squares is 4 Apply[Plus, Im[  ^Divisors[n]]] . … (d) All numbers n can be expressed as the sum of four squares, in exactly 8 Apply[Plus, Select[Divisors[n], (Mod[#, 4] ≠ 0)&]] ways, as established by Carl Jacobi in 1829.

If one starts from the single-element set {1} then applying Union , Intersection and Complement one always gets either {} or {1} . And applying Complement[s, Intersection[a, b]] to these two elements gives the same results and same equivalences as a ⊼ b applied to True and False . But if one uses instead s = {1, 2} then starts with {1} and {2} one gets any of {{}, {1}, {2}, {1, 2}} and in general with s = Range[n] one gets any of the 2 n elements in the powerset Distribute[Map[{{}, {#}} &, s], List, List, List, Join] But applying Complement[s, Intersection[a, b]] to these elements still always produces the same equivalences as with a ⊼ b .

(Similar considerations apply to the motion of quantum mechanical electrons in nested potentials.)

The particular initial condition shown can be obtained by applying the substitution system -> , -> , starting from a single black element (see page 83 ).