Generalized aggregation models One can in general have rules in which new cells can be added only at positions whose neighborhoods match specific templates (compare page 213 ). … If one puts conditions on where cells can be added one can in principle get clusters where no further growth is possible. … The second one is the analog of the system from page 331 .

The one remaining pattern is, however, much more complicated.

In each part of the proof each line can be obtained from the previous one just as on page 775 by applying the axiom or lemma indicated.

In all cases things are set up so that several steps in one rule emulate a single step in another. … This lack of dependence makes it somewhat inevitable that the only rules that end up being emulated in this way are ones with very simple behavior. … So this means that one can consider encodings based on blocks that have a kind of staircase shape—as in the rule 45 example shown.

Empirical metamathematics One can imagine a network representing some field of mathematics, with nodes corresponding to theorems and connections corresponding to proofs, gradually getting filled in as the field develops. Typical rough characterizations of mathematical results—independent of their detailed history—might then end up being something like the following: • lemma: short theorem appearing at an intermediate stage in a proof • corollary: theorem connected by a short proof to an existing theorem • easy theorem: theorem having a short proof • difficult theorem: theorem having a long proof • elegant theorem: theorem whose statement is short and somewhat unique • interesting theorem (see page 817 ): theorem that cannot readily be deduced from earlier theorems, but is well connected • boring theorem: theorem for which there are many others very much like it • useful theorem: theorem that leads to many new ones • powerful theorem: theorem that substantially reduces the lengths of proofs needed for many others • surprising theorem: theorem that appears in an otherwise sparse part of the network of theorems • deep theorem: theorem that connects components of the network of theorems that are otherwise far away • important theorem: theorem that allows a broad new area of the network to be reached. … The theorem with the longest proof is the one that states that there are only five Platonic solids.

One might think that adding rules to a system could never reduce its computational sophistication. And this is correct if with suitable input one can always avoid the new rules. … And indeed this happens when one goes from ordinary group theory to commutative group theory, and from general field theory to real algebra.

And with this setup, t steps of evolution can be found with SSSEvolveList[rule_, init_s, t_Integer] := NestList[(# /. rule)&, init, t] Note that as an alternative to having s be Flat , one can explicitly set up rules based on patterns such as s[x___, 1, 0, y___]  s[x, 0, 1, 0, y] . And by using rules such as s[x___, 1, 0, y___]  {s[x, 0, 1, 0, y], Length[s[x]]} one can keep track of the positions at which substitutions are made. ( StringReplace replaces all occurrences of a given substring, not just the first one, so cannot be used directly as an alternative to having a flat function.)

General rules [for multidimensional cellular automata] One can specify the neighborhood for any rule in any dimension by giving a list of the offsets for the cells used to update a given cell. … One can specify a neighborhood configuration by giving in the same order as the offset list the color of each cell in the neighborhood. … If a cellular automaton rule takes the new color of a cell with neighborhood configuration IntegerDigits[i, k, Length[os]] to be u 〚 i + 1 〛 , then one can define its rule number to be FromDigits[Reverse[u], k] .

Multidimensional multiway systems As a generalization of multiway systems based on 1D strings one can consider systems in which rules operate on arbitrary blocks of elements in an array in any number of dimensions.

But if one identifies time with position down the page, the presence of connections that go up as well as down the page implies that in some sense time does not always progress in the same direction. Yet at least in the cases shown here there is still a strong average flow down the page—agreeing with our everyday perception that time progresses only in one direction.