There is a very simple “Iron Rule” of cellular automaton behavior: more local growth = more global growth. Generally, adding additional birth or survival conditions increases the “aggressiveness” of the automaton. This means most or all of at least the following parameters:
- increases the growth rate of exploding CA
(decreases the decay rate of stabilizing CA); - increases seed lifetime or likelihood of seed explosion;
- increases the equilibrium density (whether ash, glass or expanding chaos).
Going from any given random rule like B356/S247 to a neighboring one like B356/S2467, it is thus immediately clear what kind of a change is expected. In this case it turns out the addition of S6 is sufficient for turning a relatively fast-stabilizing “class 2” rule into a relatively slowly-exploding “class 3” one. And the same relationship holds further also with finer, non-totalistic detail in transition rules. Using Golly’s notation, if we subtract the S6k condition, the rule remains explosive but grows now at only maybe 40% the same rate. If we subtract also S6a, we will be right back to decaying rather than growing; if we then add also B4e, the decay rate will be slower; etc.
A lot of the typology of cellular automata could be outlined already by surveying the relative strengths of each individual transition condition. E.g. as is known to most aficionados of life-like CA, the condition B2 is alone sufficient to ensure “class 3” overall exploding behavior; or, as is known to people studying also the finer non-totalistic detail, this fact results purely from the life-speed “domino engine” produced by the B2a transition; and upon its subtraction, plenty of B2-a rules will prove to be stabilizing instead. (My username here on WordPress comes from making this observation in general form back in 2012, though it had been nascently there already in the context of the Just Friends rule explored since 2000.)
The Iron Rule is however not actually watertight. E.g. experiments starting from Conway’s Life, right on the boundary of chaos, will indeed show rapid explosion for most “just upward” rules like B34/S23 or B3/S236, or slightly slower explosion for e.g. B3/S237. And the three “just downward” rules — B/S23, B3/S2, B3/S3 — are all in turn strongly stabilizing. For some reason or the other, however, the rule B35/S23 (it seems to have been named “Grounded Life” since I last checked) quite prominently backtracks on a few metrics. Sure enough, the rule still creates denser puffs of chaos, but at the same time they stabilize faster, and to sparser ash, than does regular Life. Some of this can be, it seems, attributed to a balance between stable vs. oscillating ash, as B5 suffices to destabilize most still lifes from Life; and all of this also seems to be linked to the kind of unusual “many-cloud” texture of Life and many of its immediate neighbors, which may be itself a topic worth exploring on this blog eventually.
I find the most spectacular failure of the Iron Rule to be however the rule B345/S456 — no common name that I know of, but the largely similar B345/S4567 has been called “Assimilation“. (I could call them respectively perhaps “Low Assimilation” and “High Assimilation“.) Three of the six only weakly downward neighbor rules: B34/S456, B345/S45, B345/S46 (all likewise with the addition of S7) already form explosive messes, the first one outputting stable glass (density about 69%), the latter two active chaos (densities around 52%–51%). This rule however stabilizes fast into diamond patterns with a mostly stable glassy interior (density slightly below 72%), surrounded by a mostly one-cell-thin period-2 boundary. Furthermore, despite extruding orthogonal growth with ease, the rule seems to be nearly incapable of diagonal growth! Most starting soups, even quite large, will not grow more than a few cells beyond their initial bounding diamond before stabilizing.
An about 250 by 250 stabilized diamond in B345/S456
The appearence of a stable interior is not a mystery; after all it’s already there in B34/S456, and the weaker S45 or S46 are simply not sufficient to support any stable glass texture (even relatively few agars that would be stable under B45). By the addition of birth rules that disrupt this texture (B6, B7, B8) we would also obtain chaotic interiors again, which suffice to also generate explosive growth again. Likewise, the bimodal texture as such is also not a problem. All glass-producing rules must follow something of the sort, with boundary formation distinct from the interior. In the absense of S0–3, it also easily follows that the boundary region must be chaotic or oscillating rather than stable. It is also not too difficult to prove that the typical makeup of the resulting P2 boundary is, indeed, stable.
The mystery is, rather, why does this boundary tend towards a thickness of 1 along all edges, especially diagonal? (Pattern corners may show localized larger patches, often with higher periodicity — my example pattern here in fact has a p16 upper tip with periodic bounding box decay.) There is no general rule about this for B345 rules lacking low survival conditions. This is well demonstrated in one form already by the growing chaos of B34/S456; in another by the twice downwards neighbor B345/5 (a.k.a. LongLife), which forms large oscillating phoenix regions with ease (many, but by no means all, of them with similar p2 boundary formation); and in a third form, by the fact that soups can still show just a cell or two of initial diagonal growth. Indeed, upward neighboring chaotic rules, say B3458/S456, still maintain the diamond shape, and grow mainly due to collisions of diamond-internal patches of chaos with the boundary, which will trigger occasional diagonal growth events (which then propagate at lightspeed down the diamond edges).
An attempt to sketch an explanation could perhaps begin from considering the Assimilation rules as supporting two distinct live phases — the stable glass phase and a less dense dynamic boundary phase. We can then argue that the former arises easily within the latter, and also grows rapidly within it, hence generally all the way out to the boundary of the pattern in general. But why should this process lead to the boundary stabilizing at p2? Why does the presence of the glass phase not perturb the dynamic phase to show additional diagonal growth? It seems this should be possible: this is shown e.g. by the just slightly downward non-totalistic neighbor rule B345/S4-q56, which produces not just similar glassy interior but also many identical thin p2 boundaries. And yet, enough pockets of boundary chaos remain in it, such that larger seeds will show slow explosive diagonal growth as well. Worse yet, how would the specifically diamondoid growth habit drop out of any consideration of this sort, which seems to predict nothing about a particular shape?
Should we perhaps try a deeper dive to diagnose not two, but three phases — both a semi-chaotic phoenix phase and a denser true chaotic phase? This is testable; batches of the phoenix phase can be generated easily by running soups in LongLife (or also e.g. B345/S56), and switching them over to Low Assimilation after they’ve stabilized “in texture”. Glass phase will then arise rapidly at boundaries of differently-phased phoenix phases (if you’ll excuse slightly overloading the term “phase”) or, in their absense, from hand-placed seeds. Observing this process, it is not clear to me at all however if there is much of a “true chaos phase” active at their boundary. If there is one, it does not seem to spread into the phoenix phase much faster at all than the glass phase does. Or does the glass phase actually expand, or is it simply repeatedly re-seeded by the chaos phase itself? Perhaps this angle of approach will require first of all some ability to actually delineate phases better than can be done just by visual inspection … already itself an interesting question, perhaps tractable e.g. by custom rules with sub-colorization, but probably requiring further tools and theoretical development.
Cellular Cosmology 