With a list of length n , Nest[NLFSRStep[f, taps, #] &, list, n] gives one step in the evolution of the cellular automaton in a register of width n , with a certain kind of spiral boundary condition. … One set of computations concerned functions f[{w_, x_, y_, z_}] := Mod[w + y + z + x y + x z + y z, 2] (apparently chosen to have balance between 0's and 1's that would minimize correlations).

Nand expressions If one allows a depth of at most 2n any n -input Boolean function can be obtained just by combining 2-input Nand functions. … (Note that unless one introduces an explicit copy operation—or adds variables as in the previous note—there is no way to use the same intermediate result multiple times without recomputing it.)

Thus, for example, rule 30 can be given as {{1, 1, 1}  0, {1, 1, 0}  0, {1, 0, 1}  0, {1, 0, 0}  1, {0, 1, 1}  1, {0, 1, 0}  1, {0, 0, 1}  1, {0, 0, 0}  0} To use rules in this form, CAStep can be rewritten as CAStep[rule_, a_List] := Transpose[{RotateRight[a], a, RotateLeft[a]}] /. rule or CAStep[rule_, a_List] := Partition[a, 3, 1, 2] /. rule The rules that are given can now contain patterns, so that rule 90, for example, can be written as {{1, _, 1}  0, {1, _, 0}  1, {0, _, 1}  1, {0, _, 0}  0} But how can one set up a program that can handle rules in several different forms? … Then, for example, one can define CAStep[ElementaryCARule[rule_List], a_List] := rule 〚 8 - (RotateLeft[a] + 2 (a + 2 RotateRight[a])) 〛 CAStep[GeneralCARule[rule_, r_Integer:1], a_List] := Partition[a, 2r + 1, 1, r + 1] /. rule CAStep[FunctionCARule[f_, r_Integer:1], a_List] := Map[f, Partition[a, 2r + 1, 1, r + 1]] Note that the second two definitions have been generalized to allow rules that involve r neighbors on each side.

Most of these models have far more parameters than the simple one used here, and by varying these parameters it is almost always possible to get forms that probably do not correspond to real shells. … One widespread issue concerns the orientation of the opening to a shell.

One can count the number of occurrences of each of the k b possible blocks of length b in a given state using BC[list_] := With[{z = Map[FromDigits[#, k] &, Partition[list, b, 1, 1]]}, Map[Count[z, #] &, Range[0, k b - 1]]] Conserved quantities of the kind discussed here are then of the form q . … (With k = 2 , for b = 1 , {1, 1} represents conservation of the total number of cells, regardless of color, while for b = 2 , {1, 1, 1, 1} represents the same thing, while {0, 1, -1, 0} represents the fact that in going along in any state the number of black-to-white transitions must equal the number of white-to-black ones.)

In CDMA technology for cellular telephones, for example, data is overlaid on LFSR sequences, and sequences other than the one intended for a particular receiver seem like noise which can be ignored.

Note that the explicit names of ("bound") variables in such pure functions are never significant—which is why in Mathematica one can for example use s = (# + 1) & .

Rule 170 is the classic shift map which shifts all cell values one position to the left without changing them.

Digital slope representation One can approximate a line of any slope h as in the picture below by a sequence of segments on a square grid (such as a digital display device).

Nested radicals Given a list of integers acting like digits one can consider representing numbers in the form Fold[Sqrt[#1 + #2]&, 0, Reverse[list]] .