In all cases, the output begins with a preamble specifying which block is to be represented by which codeword.

For having once specified the basic sequence to be repeated, all that then needs to be given is a pointer to this sequence, together with a representation of the total length of the data.

So if we listen to nested sequences, for example, we have no direct way to tell that they are nested, and indeed all we seem sensitive to are some rather simple features of the spectrum of frequencies that occur.

But with rule 30 most sequences that are generated—even from simple initial conditions—appear completely random with respect to all of the methods of perception and analysis discussed so far.

Indeed, considering all the kinds of systems that we have found can exhibit universality, it would have been quite peculiar if arithmetic had somehow not been able to support it.

With 3 states and 2 colors it turns out that with blank initial conditions all of the 2,985,984 possible Turing machines of this type quickly evolve to produce simple repetitive or nested behavior.

But in almost all cases, one imagines constructing examples to perform particular tasks, with a huge number of possible states and a huge number of possible colors for each cell.

What I do here is simply to divide the whole width of the picture equally among all elements that appear at each step.

Mathematica is available from Wolfram Research for all standard computer systems; much more information about it can be found on the web, especially from www.wolfram.com . … They should run without any change in all subsequent versions of Mathematica, and the majority will also run in prior versions, all the way back to Mathematica 1 (released 1988) or Mathematica 2 (released 1990). Most of the programs require only the language component of Mathematica—and not its mathematical knowledge base—and so should run in all software systems powered by Mathematica, in which language capabilities are enabled.

And particularly if there are few colors arranged in ways that are not visually familiar it is typical for this effect to make all sorts of spurious patterns appear.