The input is an integer that gives a position in either of the two rows of cells at the bottom of each picture. … Each row above the bottom one corresponds in effect to a successive register machine—and shows, if relevant, its output when given as input the integer corresponding to that position in the row, together with the complete bottom row of cells found so far.

The scheme for numbering rules works so that if the value of a particular cell is q , the value of its left neighbor is p , and the value of its right neighbor is r , then the element at position 8 - (r + 2(q + 2p)) in the list obtained from ElementaryRule will give the new value of the cell.

And by using rules such as s[x___, 1, 0, y___]  {s[x, 0, 1, 0, y], Length[s[x]]} one can keep track of the positions at which substitutions are made. ( StringReplace replaces all occurrences of a given substring, not just the first one, so cannot be used directly as an alternative to having a flat function.)

There is also exchange of DNA between paternal and maternal chromosomes, typically with a few crossovers per chromosome, at positions that seem more or less randomly distributed among many possibilities (the details affect regions of repeating DNA used for example in DNA fingerprinting).

Implementation [of Turing machines] The state of a Turing machine at a particular step can be represented by the triple {s, list, n} , where s gives the state of the head, list gives the values of the cells, and n specifies the position of the head (the cell under the head thus has value list 〚 n 〛 ).

Ulam systems Having formulated the system around 1960, Stanislaw Ulam and collaborators (see page 877 ) in 1967 simulated 120 steps of the process shown below, with black cells after t steps occurring at positions Map[First, First[Nest[UStep[p[q[r[#1], #2]] &, {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}, #] &, ({#, #} &)[{{{0, 0}, {0, 0}}}], t]]] UStep[f_, os_, {a_, b_}] := {Join[a, #], #} &[f[Flatten[ Outer[{#1 + #2, #1} &, Map[First, b], os, 1], 1], a]] r[c_]:= Map[First, Select[Split[Sort[c], First[#1]  First[#2] &], Length[#]  1 &]] q[c_, a_] := Select[c, Apply[And, Map[Function[u, qq[#1, u, a]], a]] &] p[c_]:= Select[c, Apply[And, Map[Function[u, pp[#1, u]], c]] &] pp[{x_, u_}, {y_, v_}] := Max[Abs[x - y]] > 1 || u  v qq[{x_, u_}, {y_, v_}, a_] := x  y || Max[Abs[x - y]] > 1 || u  y || First[Cases[a, {u, z_}  z]]  y These rules are fairly complicated, and involve more history than ordinary cellular automata.

Then for the case of rule 60 with n cells and cyclic boundary conditions, the state obtained after t steps is given by PolynomialMod[(1 + x) t z, {x n - 1, 2}] where z is the polynomial representing the initial state, and z = 1 for a single black cell in the first position.

But so long as there is effective randomness in the successive positions of these cells, and so long as the total number of them is conserved, then it appears that DLA-like results are usually obtained.

Largely from this grew the field of morphometrics, in which the relative positions of features such as eyes or tips of fins are compared in different species.

In recent years it has become clear that the origin of this phenomenon is that beyond the original cone cells, most color-sensitive cells in our visual system respond not to absolute color levels, but instead to differences in color levels at slightly different positions.