Endemic periodic patterns and still lifes

[confocaloid]

Two 77-cell solutions based on the p146 pi-heptomino hassler:

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#C p146 pi-heptomino hassler + DRH 3.2.1 (mini pressure cooker)
x = 39, y = 26, rule = B3/S23
33bo$32bobo$32bobo$31b2ob2o$30bo5bobo$31bobobob2o$29bobo3bo$13bo13b3ob
obobo$2o10b3o11bo5bobo$bo24b2o5bo$bobo10bo$2b2o7bo2bo$11b3o5$4b2o$5bo$
2b3o$2bo2$8b2o3bo5bo3bo$8b2o2bobo3b2o3b2o$13b2o3b2o3b2o$18bo5bo!

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#C p146 pi-heptomino hassler + DRH 4.16.1
x = 44, y = 22, rule = B3/S23
5b2o3b2o$7bobo$8bo34bo$2bo3b2ob2o3bo12bo13b3o$bob2o2bobo2b2obo10b3o11b
o$bo5bobo5bo24b2o$2o6bo6bobo10bo$16b2o7bo2bo$25b3o5$18b2o$19bo$16b3o$
16bo2$22b2o3bo5bo3bo$22b2o2bobo3b2o3b2o$27b2o3b2o3b2o$32bo5bo!

[iNoMed]

I believe this 66-cell p146 Pi hassler variant eliminates S4c from the equation, not sure if it's actually endemic though:

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x = 24, y = 32, rule = B3/S23
2bo$2b3o$5bo7b2o$4b2o7bo$11bobo$11b2o3$16b2o$16b2o2$5b2o$2bo3bo10bo$b
2o3bo9bobo$2bob2o11b2o8$22b2o$18b2obobo$18b2obo$22bo$2b2o14b2ob2o$bobo
15bobo$bo11b2o4bobo$2o11bo6bo$14b3o$16bo!

[confocaloid]

I believe this 66-cell p146 Pi hassler variant eliminates S4c from the equation, not sure if it's actually endemic though:

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x = 24, y = 32, rule = B3/S23
2bo$2b3o$5bo7b2o$4b2o7bo$11bobo$11b2o3$16b2o$16b2o2$5b2o$2bo3bo10bo$b
2o3bo9bobo$2bob2o11b2o8$22b2o$18b2obobo$18b2obo$22bo$2b2o14b2ob2o$bobo
15bobo$bo11b2o4bobo$2o11bo6bo$14b3o$16bo!

I think yes - this script says it is endemic.

Edit:

Notice that this is (roughly) equivalent to a set covering problem, where each object “covers” a certain set of transitions and costs a certain number of cells, and we want to minimize the cost (number of cells).

At least before getting sufficiently close to an optimal solution where one cell may belong to two or more recognizable sub-objects at once and a single high-period evolutionary sequence may cover most transitions.

[pzq_alex]

Notice that this is (roughly) equivalent to a set covering problem, where each object “covers” a certain set of transitions and costs a certain number of cells, and we want to minimize the cost (number of cells).

Edit:

Notice that this is (roughly) equivalent to a set covering problem, where each object “covers” a certain set of transitions and costs a certain number of cells, and we want to minimize the cost (number of cells).

At least before getting sufficiently close to an optimal solution where one cell may belong to two or more recognizable sub-objects at once and a single high-period evolutionary sequence may cover most transitions.

Yes, I agree that it is not exactly equivalent.

[carsoncheng]

Crossposting from Discord from over a month ago:
65-cell p2070 endemic pattern (p45 pi hassler on Gabriel's p138):

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x = 43, y = 49, rule = B3/S23
9b2o$9b3o$9b2o3$8b3o$3o5b3o$3o4bobo$bo7b2o$bo7b2o$bo6b3o$obo5bob2o$8b
3o4b2o$8bobo4b2o$obo6b2o$bo6b2o$bo7bo19bo$bo6b3o18bo2bo$3o$3o26bo3bo$
30bobo$6bo5b2o15b3o$5bobo3bo13bo2b2o$5b2o5b2o10bo3b2o$23b3o$13bo$11bo
2bo10b2obo$15bo10b2obo$14b2o9bo2b2o$10b2obob3o3b3o4b2o4bo$16b2o2bob2o
3bo6b2o$21bo2bo5bobobobo$20b2o9b2o$16bobobobo5bo2bo$17b2o6bo3b2obo2b2o
$18bo4b2o4b3o3b3obob2o$23b2o2bo9b2o$23bob2o10bo$24bob2o10bo2bo$39bo$
27b3o$23b2o3bo$23b2o2bo$21b3o$20bobo$19bo3bo2$20bo2bo$23bo!

Note that it still works as a p2070 oscillator in b3s234c, although the p138 has a slightly different evolutionary sequence.

[Inferno]

61-cell pattern endemic to B3/S23, which contains a p129 r-pentomino hassler with an unnamed p2 oscillator which disallows S4c.

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x = 65, y = 35, rule = B3/S23
46b2o$43b2o2bo$42bo2b2o$42b2o$46b3o$42b2obo2bo$26bo14bobobobo$24b3o15b
o3bo$23bo$23b2o$2o$bo$bobo$2b2o2$20b2o$19b2o$20bo6$44bo$44b2o$43b2o2$
61b2o$61bobo$63bo$63b2o$40b2o$41bo$38b3o$38bo!

[carsoncheng]

61-cell pattern endemic to B3/S23, which contains a p129 r-pentomino hassler with an unnamed p2 oscillator which disallows S4c.

Code: Select all

x = 65, y = 35, rule = B3/S23
46b2o$43b2o2bo$42bo2b2o$42b2o$46b3o$42b2obo2bo$26bo14bobobobo$24b3o15b
o3bo$23bo$23b2o$2o$bo$bobo$2b2o2$20b2o$19b2o$20bo6$44bo$44b2o$43b2o2$
61b2o$61bobo$63bo$63b2o$40b2o$41bo$38b3o$38bo!

3-cell reduction by welding the two oscillators, making the minpop 58:

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x = 71, y = 31, rule = B3/S23
26bo$24b3o$23bo$23b2o$2o$bo$bobo$2b2o2$20b2o$19b2o$20bo6$44bo$44b2o$43b
2o2$61b2o$61bobo$63bo4bo$63bobobobo$40b2o22b2obo2bo$41bo26b3o$38b3o23b
2o$38bo25bo2b2o$65b2o2bo$68b2o!

[AbhpzTa]

61-cell pattern endemic to B3/S23, which contains a p129 r-pentomino hassler with an unnamed p2 oscillator which disallows S4c.

Code: Select all

x = 65, y = 35, rule = B3/S23
46b2o$43b2o2bo$42bo2b2o$42b2o$46b3o$42b2obo2bo$26bo14bobobobo$24b3o15b
o3bo$23bo$23b2o$2o$bo$bobo$2b2o2$20b2o$19b2o$20bo6$44bo$44b2o$43b2o2$
61b2o$61bobo$63bo$63b2o$40b2o$41bo$38b3o$38bo!

3-cell reduction by welding the two oscillators, making the minpop 58:

Code: Select all

x = 71, y = 31, rule = B3/S23
26bo$24b3o$23bo$23b2o$2o$bo$bobo$2b2o2$20b2o$19b2o$20bo6$44bo$44b2o$43b
2o2$61b2o$61bobo$63bo4bo$63bobobobo$40b2o22b2obo2bo$41bo26b3o$38b3o23b
2o$38bo25bo2b2o$65b2o2bo$68b2o!

Minpop 56:

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x = 68, y = 31, rule = B3/S23
26bo$24b3o$23bo$23b2o$2o$bo$bobo$2b2o2$20b2o$19b2o$20bo6$44bo$44b2o$
43b2o2$61b2o$61bobo$63bo2b2o$63bobobo$40b2o19b2o2bo$41bo18bo5bo$38b3o
20b2obobo$38bo23bob2o$62bo$61b2o!

[LaundryPizza03]

Cross-posting from discord a 94-cell endemic still life found by AlbertArmStain on August 23:

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x = 24, y = 12, rule = B/S012345678
8bo$4bobo5bobobo3bo$3b2obo4bo3bo6bo$4bobobobobobob2o2b2o$2b3o2b2o6b2o
2b2o2bo$6b7o2bobo2b2obo$3bob2ob3o4bo2bob2o$13obob2o2b2o$3bo2bob2obobob
o6bo$b2o3b2o2bob2obo$bo3bo$2o2bo!

[confocaloid]

BTW the same pattern was previously posted in another thread

Cross-posting from discord a 94-cell endemic still life found by AlbertArmStain on August 23:

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x = 24, y = 12, rule = B/S012345678
8bo$4bobo5bobobo3bo$3b2obo4bo3bo6bo$4bobobobobobob2o2b2o$2b3o2b2o6b2o
2b2o2bo$6b7o2bobo2b2obo$3bob2ob3o4bo2bob2o$13obob2o2b2o$3bo2bob2obobob
o6bo$b2o3b2o2bob2obo$bo3bo$2o2bo!

122

some slight reductions to that:

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x = 24, y = 12, rule = B/S012345678
8bo$4bobo5bobobo3bo$3b2obo4bo3bo6bo$4bobobobobobob2o2b2o$2b3o2b2o6b2o2b2o2bo$
6b7o2bobo2b2obo$3bob2ob3o4bo2bob2o$13obob2o2b2o$3bo2bob2obobobo6bo$b2o3b2o2bo
b2obo$bo3bo$2o2bo!

[NimbleRogue]

What is the endemic periodic pattern with the lowest score, where the score is the sum of the total cells on for every generation of it's cycle?
For example just to be crystal clear with the counting, this pattern from life(I know it's non endemic) would get a score of 18+14+20+18+26+18+16+12=142

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x = 6, y = 6, rule = B3/S23
3o$3o$3o$3b3o$3b3o$3b3o!

For right now I think the answer is this

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x = 24, y = 12, rule = B/S012345678
8bo$4bobo5bobobo3bo$3b2obo4bo3bo6bo$4bobobobobobob2o2b2o$2b3o2b2o6b2o
2b2o2bo$6b7o2bobo2b2obo$3bob2ob3o4bo2bob2o$13obob2o2b2o$3bo2bob2obobob
o6bo$b2o3b2o2bob2obo$bo3bo$2o2bo!

Also becuase It seems likly that still lifes will always beat out oscillators I have a diffrent question. what is the endemic pattern, with a volatility of one, that has the lowest score?
Edit: I also reduced the still life's size to 89

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x = 22, y = 12, rule = B/S012345678
bobobo$3b3o$2bobo5bobobo$b2obo4bo3bo$2bobobobobobob2o$3o2b2o6b2o2b2o2b
o$o3b7o2bobo2b2obo$bob2ob3o4bo2bob2o$11obob2o2b2o$bo2bob2obobobo6bo$4b
2o2bob2obo3bo$3bo!

[toroidalet]

77-cell endemic pattern in B/S012345678:

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x = 14, y = 16, rule = B/S012345678
5bo$4bo$3b4o$4bobobo2bo$3obobo5bo$obo2bobo2bo2bo$2b3obo5b2o$obob3o4b2o
$ob3ob3ob3o$2b4obo2b2o$2b5ob2o$bobo5bo$obob3ob2o$8b3o$2b2o5bo$2b2o!

What is the endemic periodic pattern with the lowest score, where the score is the sum of the total cells on for every generation of it's cycle?

This p129 has a score of 12,363:

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x = 73, y = 30, rule = B3/S23
26bo$24b3o$23bo$23b2o$2o$bo$bobo$2b2o2$20b2o$19b2o$20bo6$44bo$44b2o$
43b2o$67bo$61b2o3bobo$61bobobobobo$63bobo3bo$63bobobobob2o$40b2o22bo5b
obo$41bo23b2ob2o$38b3o25bobo$38bo27bobo$67bo!

It seems that this other scoring method is essentially a quest to find the endemic oscillator of the lowest period; a low-period oscillator with an average population of ~1000 would be considerably better.