The diversity of these spectra is quite striking: some have simple nested forms dominated by a few isolated peaks at specific frequencies, while others have quite complex forms that cover large ranges of frequencies.

In a typical case, the initial conditions for a system like a cellular automaton can be viewed as corresponding to the input to a computation, while the state of the system after some number of steps corresponds to the output.

And while these structures may at first seem more like those in rule 54 than rule 110, I strongly suspect that the complexity of the typical behavior of rule 73 will be reflected in more sophisticated interactions between the structures—and will eventually provide what is needed to allow universality to be demonstrated in much the same way as in rule 110.

But while some of these give behavior that looks slightly more complicated in detail, as in cases (a) and (b) on the next page , all ultimately turn out to yield just repetitive or nested patterns—at least if they are started with all cells white.

The most common number of steps before halting is always n , while the maximum numbers of steps for n up to 8 is {1, 3, 5, 10, 16, 37, 215, 1280} where in the last case this is achieved by {i[1], d[2, 7], d[2, 1], i[2], i[2], d[1, 4], i[1], d[2, 3]}

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

Note that Abs[u] is plotted in the second picture, while for the last equation a common less symmetrical form replaces the last term by u[t, x] ∂ x u[t, x] .)

Many equations used in physics can lead to singularities: the Navier–Stokes equations for fluid flow yield shock waves, while the Einstein equations yield black holes.

The tetrahedron network from page 476 is for example given in this representation by {1  {2, 3, 4}, 2  {1, 3, 4}, 3  {1, 2, 4}, 4  {1, 2, 3}} The list of nodes reached by following up to n connections from node i are then given by NodeLists[g_, i_, n_] := NestList[Union[Flatten[# /. g]] &, {i}, n] The network distance corresponding to the length of the shortest path between two nodes is given by Distance[g_, {i_, j_}] := Length[NestWhileList[ Union[Flatten[# /. g]] &, {i}, !

Some of these numbers are even, while some are odd.