All of these are either trivial or essentially equivalent to rules 90 or 150. Of all k k 2r + 1 rules with k colors and range r it turns out that there are always exactly k 2r + 1 additive ones—each obtained by taking the cells in the neighborhood and adding them modulo k with weights between 0 and k - 1 .

The lengths of the longest cycles are given on page 951 ; all other cycles must have lengths which divide these. … When the number of cells n is odd this structure consists of a single arc, so that half of all states lie on cycles.

Implementation [of constraint satisfaction] The number of squares violating the constraint used here is given by Cost[list_] := Apply[Plus, Abs[list - RotateLeft[list]]] When applied to all possible patterns, this function yields a distribution with Gaussian tails, but with a sharp point in the middle. … The third curve shown on page 346 is obtained from Table[Cost[IntegerDigits[i, 2, n]], {i, 0, 2 n - 1}] There is no single ordering that makes all states which can be reached by changing a single square be adjacent.

The next integer after all of the ordinary ones—the first infinite integer—is given the name ω . … However, not all infinite integers can be represented in this way. … And it turns out that in general one in effect has to introduce an infinite sequence of names in order to be able to specify all transfinite integers.

Yet so far in this book all the systems we have discussed have effectively been limited to just one dimension.

In each case the result of 10 steps of evolution is shown, and the pictures are scaled so that all points above the bottom of the original stem can be included.

[Structures in] other 2D cellular automata The general problem of finding persistent structures is much more difficult in 2D than in 1D, and there is no completely general procedure, for example, for finding all structures of any size that have a certain repetition period.

One could in principle imagine defining mathematics to encompass all studies of abstract systems, and indeed this was in essence the definition that I had in mind when I chose the name Mathematica. … But despite all these issues, many mathematicians implicitly tend to assume that somehow mathematics as it is practiced is universal, and that any possible abstract system will be covered by some area of mathematics or another. The results of this book, however, make it quite clear that this is not the case, and that in fact traditional mathematics has reached only a tiny fraction of all the kinds of abstract systems that can in principle be studied.

And in all cases, there are considerable fluctuations in the periods that occur as the size changes. So how does all of this relate to class 2 behavior?

And as discussed above, there is good evidence that the center column of rule 30 is indeed random according to all reasonable definitions of this kind. So whether or not one chooses to say that the sequence is truly random, it is, as far as one can tell, at least random for all practical purposes.