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 object may "cover" most or all of the 102 isotropic 3-by-3 conditions.

edit 2024-11: fixed previously incorrect terminology and reworded.

[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.

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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!

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.

[NooneAtAll3]

can p2 pattern be endemic?

what's the minpop for p2 life-endemic pattern?

[hotdogPi]

In the 2024 newjslife stamp collection (https://github.com/carson-cheng/newjslife/tree/main), which contains everything that's been posted on the forum until mid-2024, all p2 oscillators work with S8. I can't quite find a simple proof that p2 oscillators in Life can't fail with S8, but given how many are in that collection, I believe it to be the case. (The converse is not true; there are some B3/S238 p2s that fail in Life.)

[vilc]

In the 2024 newjslife stamp collection (https://github.com/carson-cheng/newjslife/tree/main), which contains everything that's been posted on the forum until mid-2024, all p2 oscillators work with S8. I can't quite find a simple proof that p2 oscillators in Life can't fail with S8, but given how many are in that collection, I believe it to be the case. (The converse is not true; there are some B3/S238 p2s that fail in Life.)

Here is an example of a p2 oscillator which triggers the S8 transition, found with LLS.

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x = 14, y = 12, rule = B3/S23
4bo4bo$3bob2obobob2o$3bob3o2bobo$2obo6bobo$obo3b3o3b2o$2bobob3o2bo$bob
2ob4obo$bo7bobobo$2b4o6b2o$obo2b4o$2o3b2o$5bo!

[rabbit]

Here is an example of a p2 oscillator which triggers the S8 transition, found with LLS.

Searching for a 13x13 pattern centered around the 3x3 live box yielded 0 results with RLS, so this may be one of the smallest examples.

[Resu]

what's the minpop for p2 life-endemic pattern?

What about p3 patterns?

[Hdjensofjfnen]

Here is an example of a p2 oscillator which triggers the S8 transition, found with LLS.

Searching for a 13x13 pattern centered around the 3x3 live box yielded 0 results with RLS, so this may be one of the smallest examples.

A bit more searching with LLS confirms that this is the minimum-population p2 (in generation 0) that triggers S8 and fits inside a 17x17 box.

[whatislife]

Inspired by the S8 oscillator above, I searched for a small p2 oscillator triggering S7e:

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x = 11, y = 12, rule = B3/S23
3b2ob2o$2bobobo$2bobo2b3o$b2obo5bo$o4bob4o$b3ob3o$2bo2b4o$2bo6bo$b3obo
$o4b4o$2o3b2o$5bo!

and S7c:

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x = 13, y = 12, rule = B3/S23
7b2ob2o$6bobobo$5bobo4bo$4bob3o2b2o$3bo2bobobo$2bo5bobo$bob2ob2obo$obo
bo2b2o$b2ob2obo$3bobo$3bo4bo$2b2o3b2o!

They are not proven to be minimal, but those are the only rotors I could get out of RLS at 12x12 and 13x13 respectively.

[ElijahKen]

...
and S7c:

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x = 13, y = 12, rule = B3/S23
7b2ob2o$6bobobo$5bobo4bo$4bob3o2b2o$3bo2bobobo$2bo5bobo$bob2ob2obo$obo
bo2b2o$b2ob2obo$3bobo$3bo4bo$2b2o3b2o!

They are not proven to be minimal, but those are the only rotors I could get out of RLS at 12x12 and 13x13 respectively.

Reduced by two cells:

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x = 12, y = 13, rule = B3/S23
7bo$6bobo$5bo2bo2bo$4bo4b3o$3bob2o$2bob3ob2o$bo2b3o2bo$o7bob2o$b2ob2o
2bob2o$3b2obo$b2o2bo$bo2bo$2b2o!

Edit: Prettified RLE.
EDIT 2:
Reduced the other (S7e) example by 3 cells (49 & 49 -> 48 & 46):

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x = 12, y = 13, rule = B3/S23
6bo2b2o$4b3o2bo$3bo7bo$3bob2ob4o$4bob3o$6b4o$4bo5bo$3bo2bo$b2o3b4o$4ob
2o$3bobobo$3bo2b2o$2b2o!

[WhiteHawk]

Unsynthesizable oscillator 1 has B8, but is massive. What is the smallest p2 oscillator that triggers B8 in life? (Pedestrian Life has a very small p2 which uses B8)

Also I guess given the S7 and S8 oscillators above, what is the answer to the question of the smallest endemic P2 oscillator?

[ElijahKen]

Unsynthesizable oscillator 1 has B8, but is massive. What is the smallest p2 oscillator that triggers B8 in life? (Pedestrian Life has a very small p2 which uses B8)

Here is a smallish one (found just now with lls and jls):

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x = 11, y = 15, rule = B3/S23
4b2o$4bobob2o$b2obob2obo$o2bo$2o4b3o$4b3o2bo$2ob2ob2o2bo$o3b3o2b2o$b4o
$7bo$ob2ob2o$2ob3o$6b3o$5bo2bo$6b2o!

EDIT:
S7e reduced by two cells to 46 cells:

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x = 11, y = 12, rule = B3/S23
b2o3bob2o$bobo2b2obo$3bo$bo2bob3o$b2ob3o2bo$4b4o2bo$6bo2b2o$2b2o$bob4o
$obo2bo$obo4bo$bo4b2o!

B8 reduced to 43 cells:

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x = 12, y = 13, rule = B3/S23
2o5b2o$bo4bo2bo$bob2o$2bo2b3obo$5bobobo$2bo2b3o2bo$2b2obo$5bob2obo$5b
o2b2obo$6b2o2b2o$8bo$5bobo$5b2o!

[synperiplanar]

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?

i think it would be interesting to extend this question to non-oscillating patterns. this means that every generation up until the last isotropic transition is ruled out would count toward the score. so let's say the B-heptomino somehow ruled out its last transition at generation 15, this would be 7+8+10+12+14+18+16+19+23+19+22+21+22+21+15+20=267 score.

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x = 3, y = 4, rule = B3/S23
o$2o$b2o$2o!

it's definitely more interesting when you research on a fundamental standpoint rather than just searching a specific type of pattern for no reason at all.