Pattern phase types

[Tropylium]

Continued from another topic:

I've observed that a lot of Life patterns seem to altenate between a phase with mostly S2/underpopulated cells, and phases with mostly S3/overpopulated cells, so general phase type terminology might come useful; "empty" and "full" could work (or something like "sparse"/"dense", "open"/"close"). A third category with hardly any dying cells at all is also sometimes seen (maybe "growth" or "near-still").

Among oscillators, toad, beacon, mold and unix are perfect full/empty alternation examples, and figure 8 is almost perfect as well (generation 6 has some D0 sparks that don't allow calling it entirely full). This covers *WSS phases as well, and the Schick engine also does a near-perfect job at full/empty alternation. Your "hollow arrow" phase is, starting from the (full) bullet heptomino phase, the 2nd empty phase.

This doesn't stop at periodic patterns, however. Consider the evolution of traffic lights. Starting from this shape (I call it a "bunny", after the Rabbits precedessor "bunnies" which has two of them)

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*.*
.*.
.*.

which is "empty", the evolution goes:
• t-tetromino: a "growth" phase
• bullet heptomino: "full"
• one entirely "empty" generation
• another "empty" generation, but with two S3 cells
• 3×3 square: closest to "full" (tho with S2 rather than S3 cells)
• 3×3 diamond: entirely empty (but w/o dying cells; "empty-growth"?)
• filled diamond: entirely full
• unbounded tubber: "empty-growth"
• 4 crosses: entirely full
• and the resultant blinkers are entirely empty.

Or, honey farm. Starting from a bun, which is basically empty (but again without dying cells):

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.***
*..*
.**.

• Gen 1 is somewhat mixed, but could be said to have a "full" area in the upper right
• Gen 2 is entirely mixed
• Gen 3 is basically a growth phase
• after which it's a neat full - empty - full - empty alternation all the way to the beehives.

For a third example: B-heptomino. The actual "engine" runs full - empty - full - empty (the exhaust follows this fairly well but not entirely) all the way to gen 13, which is full rather than empty; then it's in inverse up to gen 19, which is empty rather than full. The Herschel itself is not a clean phase, but there's after that 6 generations of empty-full alternation again (well, gen 21 is more "growth" than "empty").

In general it seems empty/full alternation works the cleanest when a pattern grows at c/2 in some direction. An odd-period pattern is obviously not going to be able to pull this off…

(My "LifeAnnotated" coloration may come useful in visualizing all this — "empty" phases will look cold, with a white/cyan palette, "full" phases will look hot, with a yellow/red/magenta palette.)

I wonder if a more fine-grained analysis would help in dealing with patterns "skipping phases"; I've already estabilished the "growth" and "empty-growth" phases here. The next step might be distinguishing death-dominated and survival-dominated empty phases (like generations 18 and 23 of a B-heptomino), and similarly for full phases…

[Pooping_Alien]

L/M/HWSS have "fish" phase and "L" phase. Is this what the author meant?
Fish phase has more cells, L phase has less. But it is in L phase that M/H/OWSS have their side spark produced

[Tropylium]

L/M/HWSS have "fish" phase and "L" phase. Is this what the author meant?
Fish phase has more cells, L phase has less. But it is in L phase that M/H/OWSS have their side spark produced

Yes, that's correct; as I said, *WSS phases are a perfect example.

To summarize:

empty phase:
—less cells
—sparser pattern
—most dying cells are underpopulated (sparks or other edge cells)
—most surviving cells have 2 neighbors

full phase:
—more cells
—denser pattern
—most dying cells are overpopulated (at the middle of pattern)
—most surviving cells have 3 neighbors