Going beyond ordinary networks, one can consider hypernetworks in which connections join not just pairs of nodes, but larger numbers of nodes.

Some sections of chapters (usually later ones) were added well after the rest.

It should be pointed out, however, that it is wrong to think that once one has described everything as, say, computation, then there is nothing more to do.

Rule 30 inversion The total numbers of sequences for t from 1 to 15 not yielding stripes of heights 1 and 2 are respectively {1, 2, 2, 3, 3, 6, 6, 10, 16, 31, 52, 99, 165, 260} {2, 5, 8, 14, 23, 40, 66, 111, 182, 316, 540, 921, 1530, 2543, 4122} The sideways evolution of rule 30 discussed on page 601 implies that if one fills cells from the left rather than the right then some sequence of length t + 1 will always yield any given stripe of height t .

Undecidability in cellular automata For 1D cellular automata, almost all questions about ultimate limiting behavior are undecidable, even ones that ask about average properties such as density and entropy.

Implementation of digit sequences A whole number n can be converted to a sequence of digits in base k using IntegerDigits[n,k] or (see also page 1094 ) Reverse[Mod[NestWhileList[Floor[#/k] &, n, # ≥ k &], k]] and from a sequence of digits using FromDigits[list,k] or Fold[k #1 + #2 &, 0, list] For a number x between 0 and 1, the first m digits in its digit sequence in base k are given by RealDigits[x, k, m] or Floor[k NestList[Mod[k #, 1]&, x, m - 1]] and from these digits one can reconstruct an approximation to the number using FromDigits[{list, 0}, k] or Fold[#1/k + #2 &, 0, Reverse[list]]/k

But at a mathematical level one can simply ask whether a particular equation always has solutions which are at least as regular as its initial conditions.

[State networks for] shift rules The pictures below show networks obtained with rule 170, which just shifts every configuration one position to the left at each step.

But in the early 1980s, simulated annealing was suggested by Scott Kirkpatrick and others as one of the first potentially general approaches.

But in the end, after running programs for a total of several years of computer time—corresponding to more than a million billion logical operations—and creating the equivalent of tens of thousands of pages of pictures, I was finally able to find all of the various examples shown in this chapter and the ones that follow.