Pattern phase types
Posted: March 9th, 2012, 10:49 am
Continued from another topic:
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):
• 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…
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)Tropylium wrote: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.
Code: Select all
*.*
.*.
.*.• 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):
Code: Select all
.***
*..*
.**.• 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…