But in this chapter (as well as some of the ones that follow) the systems I consider have often had huge amounts written about them before, making any kind of complete summary quite impossible.

One might think that as such capabilities increase, data compression would become less relevant.

Recursive subdivision [encoding] In one dimension, encoding can be done using Subdivide[a_] := Flatten[ If[Length[a]  2, a, If[Apply[SameQ, a], {1,First[a]}, {0, Map[Subdivide, Partition[a, Length[a]/2]]}]]] In n dimensions, it can be done using Subdivide[a_, n_] := With[{s = Table[1, {n}]}, Flatten[ If[Dimensions[a]  2s, a, If[Apply[SameQ, Flatten[a]], {1, First[Flatten[a]]}, {0, Map[Subdivide[#, n] &, Partition[a, 1/2Length[a] s], {n}]}]]]]

One can also consider textures associated, say, with surface roughness of physical objects.

However, if one uses the function to generate a score—say playing a note at the position of each peak—then no such simplicity can be recognized.

[Structures in] rule 41 Various rules like rule 41 below can perhaps be viewed as having localized structures—though ones that apparently always travel in the same direction at the same speed.

And in my experience many of the intellectually most interesting aspects of technology emerge only when one actually tries to build technology for real—and they are often in a sense best captured by the technology itself rather than by a book about it.

Properties [of example multiway systems] The second rule shown has the property that black elements always appear before white, so that strings can be specified just by the number of elements of each color that they contain—making the rule one of the sorted type discussed on page 937 , based on the difference vector {{2,-1}, {-1,3}, {-4,-1}} .

Numbering scheme [for Turing machines] One can number Turing machines and get their rules using Flatten[MapIndexed[{1, -1} #2 + {0, k}  {1, 1, 2} Mod[Quotient[#1, {2k, 2, 1}], {s, k, 2}] + {1, 0, -1} &, Partition[IntegerDigits[n, 2 s k, s k], k], {2}]] The examples on page 79 have numbers 3024, 982, 925, 1971, 2506 and 1953.

For most types of systems (such as Turing machines) such non-deterministic versions do not ultimately allow any greater range of computations to be performed than deterministic ones.