So from this it was concluded that only certain supposedly perfect objects like the heavenly bodies could be expected to be fully amenable to human understanding.

Extended instruction sets [for register machines] One can consider also including instructions such as RMExecute[eq[r1_, r2_, m_], {n_, list_}] := If[list 〚 r1 〛  list 〚 r2 〛 , {m, list}, {n + 1, list}] RMExecute[add[r1_, r2_], {n_, list_}] := {n + 1, ReplacePart[list, list 〚 r1 〛 + list 〚 r2 〛 , r1]} RMExecute[jmp[r1_], {n_, list_}] := {list 〚 r1 〛 , list} Note that by being able to add and subtract only 1 at each step, the register machines shown in the main text necessarily operate quite slowly: they always take at least n steps to build up a number of size n .

However, in the cases that are analyzed in dynamical systems theory, only shifts and other very simple operations are typically performed on digit sequences.

Iterated run-length encoding Starting say with {1} consider repeatedly replacing list by (see page 1070 ) Flatten[Map[{Length[#], First[#]} &, Split[list]]] The resulting sequences contain only the numbers 1, 2 and 3, but otherwise at first appear fairly random. However, as noticed by John Conway around 1986, the sequences can actually be obtained by a neighbor-independent substitution system, acting on 92 subsequences, with rules such as {3, 1, 1, 3, 3, 2, 2, 1, 1, 3}  {{1, 3, 2}, {1, 2, 3, 2, 2, 2, 1, 1, 3}} .

In most patterns generated from initial conditions containing say a single black cell most cells whose values are not forced to be the same end up being at least slightly different—even in cases like a = 0.375 . Note that in cases like a = 0.475 there is some trace of a pattern at every step—but it only becomes obvious when it makes values wrap around from 1 to 0. The pictures below show successive colors of (a) the background (compare page 950 ) and (b) the center cell for each a = n/500 from 0 to 1 for the systems on page 159 .

A typical result is that correlations between colors of different cells fall off like a power of distance—with the specific power depending only on general features of the nested patterns formed, and not on most details of the system.

In the pictures below the liquid is divided into cells, with each cell having a temperature from 0 to 1, corresponding exactly to a continuous cellular automaton of the kind discussed on page 155 . … If the temperature of any cell exceeds 1, then only the fractional part is kept, as in the systems on page 158 , representing the consumption of latent heat in the boiling process.

Animals like starfish often have 5-fold symmetry; higher animals usually only 2-fold symmetry.

Implementation [of generalized substitution systems] Sequential substitution systems in which only one replacement is ever done at each step can just be implemented using /. as described on page 893 . Substitution systems in which all replacements are done that are found to fit in a left-to-right scan can be implemented as follows GSSEvolveList[rule_, s_, n_] := NestList[GSSStep[rule, #] &, s, n] GSSStep[rule_, s_] := g[rule, s, f[StringPosition[s, Map[First, rule]]]] f[{ }] = { }; f[s_] := Fold[If[Last[Last[#1]] ≥ First[#2], #1, Append[#1, #2]]&, {First[s]}, Rest[s]] g[rule_, s_, { }] := s; g[rule_, s_, pos_] := StringReplacePart[ s, Map[StringTake[s, #] &, pos] /. rule, pos] with rules given as {"ABA"  "BAAB", "BBBB"  "AA"} .

(In 2D it is straightforward to decide equivalence, and in 3D it is known that only some fundamental groups can be obtained—roughly because not all networks can be embedded in 2D—and it is expected that it will ultimately be possible to decide equivalence.)