Which Life-like CA have been proven omniperiodic?

[lukebradford]

I'm curious which Life-like cellular automata have been proven to be omniperiodic. Is there a class of CA for which this proof is simple? (All it says on the wiki is that Dean Hickerson proved "several rules" to be omniperiodic by searching for usable signals. http://www.conwaylife.com/wiki/Omniperiodic)

[May13]

I'm curious which Life-like cellular automata have been proven to be omniperiodic. Is there a class of CA for which this proof is simple? (All it says on the wiki is that Dean Hickerson proved "several rules" to be omniperiodic by searching for usable signals. http://www.conwaylife.com/wiki/Omniperiodic)

I was also interested in this question and did a little research.
I found three interesting posts about the omniperiodicity of several rules:

• Dean Hickerson proved that B3/S0123 is omniperiodic
• Matthias Merzenich found several signal loops in B3/S234
• I proved that B3/S123 is omniperiodic

About the second post from this list: Merzenich's signal loop from edit 2 have repeat time 7 and works in rules B3/S234 to B38/S023478. Oscillators with periods 1-6 exists in any of these rules:

Code: Select all

x = 182, y = 274, rule = B38/S023478
3bo26b2o$30b2o2$o2bo3$3bo3$3bo3$o2bo2bo8$o2bo2bo$30b3o2$6bo3$o2bo2bo3$
o3$o2bo2bo8$o2bo2bo24b2ob2o$30bobobo$30bo4bo$6bo26b2o$31b2ob2o2$o2bo2b
o3$6bo3$o2bo2bo8$o5bo27b2o$30bobo2bo$30b3o$o5bo27bo$30b2obo$30b2ob2o$o
2bo2bo3$6bo3$6bo8$o2bo2bo26b2o$33bobo$35bo$o29b2o3b2o$30bo$31b3ob2o$o
2bo2bo26bob2o3$6bo3$o2bo2bo8$o2bo2bo$30b2o3b2o$31bo3bo$o30b5o$31bo3bo$
30b2o3b2o$o2bo2bo3$o5bo3$o2bo2bo8$o2bo2bo96b2o$103bo$105bob2o$6bo8bo
88b2ob2o2bo$103bobo3b3o$102bobob3o$6bo5bo2bo2bo83b2ob2obob4o$104bo3bo
5bob2o$97b2o3b2obob2obo2b2obobo$6bo8bo81bo2b3obo4bo5bobobo$96b2ob2o5b
3o3bobobobo$99bo5b2o2b4obobo$6bo92bo2b2o5bo4b2obo$98b2obob4o5b3o2b3o$
93b2o2bo5bo2bo6bo3bo2bo$93bo2bobobobob2o5bobo5b2o$94b2ob2obobobo6b2o$
95bobo2bobob2o9b2obob3o$95bo3bobo14b2ob2o2bo$92b2ob2o3bo13bo2bo4b2o$
91bobob2o2b2obo8bo2b2obo3bo$91bo3bo3bo2b3o6b3obo2b2ob5o$88b2ob2o3b3ob
3o2bo4b2o4b2ob2ob2o2bo$87bobo2bo2b2obobo3b2o14bo4b2o$87bo3bo3bo2bo21bo
$84b2obo4b3o2b2o22b2obob3o$83bobo2bo2b2o25b3ob2ob2o2bo$83bo3bo3bo26bo
4bo2b4o$80b2ob2o3b3obo30bo2b2o$79bobo2bo2b2o3bo31b2obob3o$79bob3o3bo4b
2o27b3ob2ob2o2bo$76b2ob4ob3obo32bo4bo4b2o$75bobo2bo2b2o3bo37bo$75bo3bo
3bo4b2o37b2obob3o$72b2ob2o3b3obo39b3ob2obo3bo$71bobob2o2b2o3bo39bo4bo
4b2o$71bo2b2o3bo4b2o43bo$68b2ob2o3b3obo49b2obob3o$67bobo2bo2b2o3bo46b
3ob2ob2o2bo$67bo3bo3bo4b2o45bo4bo4b2o$64b2ob2o3b3obo55bo$63bobob2o2b2o
3bo56b2ob5o$63bo3bo3bo4b2o52b3ob2ob2o2bo$60b2ob2o3b3obo57bo4bo4b2o$59b
obo2bo2b2o3bo62bo$59bo3bo3bo4b2o62b2obob3o$56b2obo4b3obo64b3ob2ob2o2bo
$55bobo2bo2b2o3bo64bo4bo4b2o$55bo3bo3bo4b2o68bo3bo$52b2ob2o3b3obo74b2o
b5o$51bobo2bo2b2o3bo71b3ob2ob2o2bo$51bob3o3bo4b2o70bo4bo4b2o$48b2ob4ob
3obo80bo$47bobo2bo2b2o3bo81b2obob3o$47bo3bo3bo4b2o77b3ob2ob2o2bo$44b2o
b2o3b3obo82bo4bo2b4o$39bo3bobob2o2b2o3bo87bo2b2o$38bobo2bo2b2o3bo4b2o
87b2obob3o$37bobo2b3o3b3obo89b3ob2ob2o2bo$37b2ob2o2bo2b2o3bo89bo4bo4b
2o$39bo2bo4bo2b2o95bo$37b2o2b3o5b2o97b2obob3o$36bobobo2b3o5bo93b3ob2ob
o3bo$36bo3b2obo2bo4bo93bo4bo4b2o$33b2ob2o3bo3b2o3b3o97bo$34bo6bo10bo
98b2obob3o$34bobob2ob2o105b3ob2ob2o2bo$32b2obo4bo107bo4bo4b2o$32b2obo
2bobo112bo$35b2obob2ob2o109b2ob5o$32b3obobo2bobobobo4b2o97b3ob2ob2o2bo
$30bo2bobobobo3bob3o3bobo97bo4bo4b2o3b2o$30b2o2bobobo4b2o6bo104bo8bobo
$35b2ob2o2bo2b3o2b3o104b2obob4o$39bo4bo3bo3bo105b2ob2o3bo4bo$39b2o2b5o
b2ob2o102bo2bo4b2obo3b3o$40b4o3b2ob2o104b4o3bob2obo4bo$38bo4bob2obo3b
4o104b2ob2o3b4o$38b3o3bob2o4bo2bo102b2ob2ob5o2b2o$40bo4bo3b2ob2o105bo
3bo3bo4bo$46b4obob2o104b3o2b3o2bo2b2ob2o$44bobo8bo104bo6b2o4bobobo2b2o
$44b2o3b2o4bo4bo97bobo3b3obo3bobobobo2bo$49bo2b2ob2ob3o97b2o4bobobobo
2bobob3o$50b5ob2o109b2ob2obob2o$58bo112bobo2bob2o$52b2o4bo4bo107bo4bob
2o$52bo2b2ob2ob3o105b2ob2obobo$53b3obob2o98bo10bo6bo$61bo97b3o3b2o3bo
3b2ob2o$55b2o4bo4bo93bo4bo2bob2o3bo$55bo3bob2ob3o93bo5b3o2bobobo$56b3o
bob2o97b2o5b3o2b2o$64bo95b2o2bo4bo2bo$58b2o4bo4bo89bo3b2o2bo2b2ob2o$
58bo2b2ob2ob3o89bob3o3b3o2bobo$59b3obob2o87b2o4bo3b2o2bo2bobo$63b2o2bo
87bo3b2o2b2obobo3bo$61b4o2bo4bo82bob3o3b2ob2o$61bo2b2ob2ob3o77b2o4bo3b
o3bo$62b3obob2o81bo3b2o2bo2bobo$70bo80bob3ob4ob2o$64b2o4bo4bo70b2o4bo
3b3obo$64bo2b2ob2ob3o71bo3b2o2bo2bobo$65b5ob2o74bob3o3b2ob2o$69bo3bo
68b2o4bo3bo3bo$67b2o4bo4bo64bo3b2o2bo2bobo$67bo2b2ob2ob3o64bob3o4bob2o
$68b3obob2o62b2o4bo3bo3bo$76bo62bo3b2o2bo2bobo$70b2o4bo4bo57bob3o3b2ob
2o$70bo2b2ob2ob3o52b2o4bo3bo3bo$71b5ob2o56bo3b2o2b2obobo$79bo55bob3o3b
2ob2o$73b2o4bo4bo45b2o4bo3bo3bo$73bo2b2ob2ob3o46bo3b2o2bo2bobo$74b3obo
b2o49bob3o3b2ob2o$82bo43b2o4bo3b2o2bo$76b2o4bo4bo39bo3b2o2b2obobo$76bo
3bob2ob3o39bob3o3b2ob2o$77b3obob2o37b2o4bo3bo3bo$85bo37bo3b2o2bo2bobo$
79b2o4bo4bo32bob3ob4ob2o$79bo2b2ob2ob3o27b2o4bo3b3obo$80b3obob2o31bo3b
2o2bo2bobo$84b2o2bo30bob3o3b2ob2o$82b4o2bo4bo26bo3bo3bo$82bo2b2ob2ob3o
25b2o2bo2bobo$83b3obob2o22b2o2b3o4bob2o$91bo21bo2bo3bo3bo$85b2o4bo14b
2o3bobob2o2bo2bobo$85bo2b2ob2ob2o4b2o4bo2b3ob3o3b2ob2o$86b5ob2o2bob3o
6b3o2bo3bo3bo$90bo3bob2o2bo8bob2o2b2obobo$88b2o4bo2bo13bo3b2ob2o$88bo
2b2ob2o14bobo3bo$89b3obob2o9b2obobo2bobo$99b2o6bobobob2ob2o$91b2o5bobo
5b2obobobobo2bo$91bo2bo3bo6bo2bo5bo2b2o$92b3o2b3o5b4obob2o$94bob2o4bo
5b2o2bo$95bobob4o2b2o5bo$93bobobobo3b3o5b2ob2o$92bobobo5bo4bob3o2bo$
93bobob2o2bob2obob2o3b2o$94b2obo5bo3bo$98b4obob2ob2o$103b3obobo$100b3o
3bobo$100bo2b2ob2o$103b2obo$108bo$107b2o!

So rules B3/S234 to B38/S023478 are omniperiodic.
About the third post from this list: my p7+ loop works in rules B3/S123 to B38/S0123678, and oscillators with periods 1-6 exists in any of these rules:

Code: Select all

x = 271, y = 106, rule = B38/S0123678
247bo7bo$247b3o3b3o7bo$250bo12bo$247b2o4b3o3bobob3o$247bo3bo4bo2bobo$
241bobobobo3bobobobobob5o$241bobobob3obobo3bob2obo$239b3obobobo2bo2bo
3bo3bob3o$247b3o3b2o4b2o$241b2o13bo5bob4ob2o$244bobob2o3b3o3b2obobo2bo
bo$241b4obobobo2bo6bobobo4bo$246bo12bo3bo2b3o$243b3o18b3o$242bo4bo$
238b2obobo4bo10b2ob4obobo$239bobobo2bobo11bobo2bobobo$239bobob4ob2o10b
o4bobob2o$240b2o19bo4bo$241b4o17b4o$240bo4bo15b2o$239bo4bobob2o10bobob
4ob2o$239bobo2bobobo11bobobo2bobo$238b2ob4obobo10b2obobo4bo$263bo4bo$
243b3o18b3o$242bo2b3o19bo$238b2obobo4bo10b2ob4obobo$239bobobo2bobo11bo
bo4bobo$239bobob4ob2o10bo4bobob2o$261bo4bo$242b3o18b3o$240b3o2bo$239bo
2b3obob2o10bob6ob2o$239bobo2bobobo11bobobo2bobo$238b2ob4obobo10b2obobo
4bo$263bo4bo$243b3o18b3o$242bo4bo$238b2obobo4bo10b2ob4obobo$239bobo4bo
bo11bobo4bobo$239bobob4ob2o10bo4bobob2o$261bo4bo$242b3o18b3o$240bo4bo$
239bo4bobob2o10bobob4ob2o$239bobo2bobobo11bobobo2bobo$238b2ob6obo10b2o
bob3o2bo$162bobobobo94bo2b3o$46b2o39b2o33b2o36b3obobobo36bo37b3o18b3o$
46b2o75b3o38bobob2o35bo36bo4bo$44b2o39b4o34b2obo33b2obo37bobob2o31b2ob
obo4bo10b2ob4obobo$4b2o38b2o38bo4bo34bobo34bobo3b3o29b3obo3bo31bobo4bo
bo11bobo2bobobo$84bo3bo33bobo36bobobo3bo35b2o32bobob4ob2o10bo4bobob2o$
85b2o35bobo38bobob2o32b2o6bo31bo19b3o2bo$163bobo38b6o32b3o18b3o$164b2o
34bobobo35bo4bo$200bobo2bob3o29bo4bobob2o10bobob4ob2o$204b2obo31bobo2b
obobo11bobobo2bobo$238b2ob4obobo10b2obobo4bo$246b2o15bo4bo$243b4o17b4o
$242bo4bo19b2o$238b2obobo4bo10b2ob4obobo$239bobobo2bobo11bobo2bobobo$
239bobob4ob2o10bo4bobob2o$261bo4bo$242b3o18b3o$240b3o2bo3bo12bo$239bo
4bobobo6bo2bobobob4o$239bobo2bobob2o3b3o3b2obobo$238b2ob4obo5bo13b2o$
248b2o4b2o3b3o$243b3obo3bo3bo2bo2bobobob3o$246bob2obo3bobob3obobobo$
243b5obobobobobo3bobobobo$247bobo2bo4bo3bo$243b3obobo3b3o4b2o$245bo12b
o$245bo7b3o3b3o$253bo7bo12$4bo36bo2bo2bo33bo2bo2bo33bo5bo33bo2bo2bo33b
o2bo2bo33bo2bo2bo$3bo36bo2bo2bo33bo2bo2bo33bo5bo33bo2bo2bo33bo2bo2bo
33bo2bo2bo2$bo2bo42bo39bo33bo5bo33bo39bo45bo8bo$o2bo42bo39bo33bo5bo33b
o39bo45bo8bo2$4bo36bo2bo2bo33bo2bo2bo33bo2bo2bo33bo2bo2bo33bo2bo2bo39b
o5bo2bo2bo$3bo36bo2bo2bo33bo2bo2bo33bo2bo2bo33bo2bo2bo33bo2bo2bo39bo5b
o2bo2bo2$4bo36bo45bo39bo39bo33bo5bo39bo8bo$3bo36bo45bo39bo39bo33bo5bo
39bo8bo2$bo2bo2bo33bo2bo2bo33bo2bo2bo39bo33bo2bo2bo33bo2bo2bo39bo$o2bo
2bo33bo2bo2bo33bo2bo2bo39bo33bo2bo2bo33bo2bo2bo39bo!

So rules B3/S123 to B38/S0123678 are omniperiodic.
Today I discovered a signal loop with repeat time 7 in rules B36/S123:B368/S012378:

Code: Select all

x = 160, y = 266, rule = B36/S123Super
28.2A26.2A26.2pA26.2A26.2A$27.A.A25.A.A25.pA.pA25.A.A25.A.A$20.A4.A
22.A4.A22.pA4.pA22.A4.A22.A4.A$20.A5.3A19.A5.3A19.pA5.3pA19.A5.3A19.A
5.3A$17.2A.A.A6.A10.2A.A.2A.A.A6.A10.2pA.pA.2pA.pA.pA6.pA10.2A.A.2A.A
.A6.A10.2A.A.2A.A.A6.A$17.A2.A.A2.A.A13.A.A.A2.A.A2.A.A13.pA.pA.pA2.pA
.pA2.pA.pA13.A.A.A2.A.A2.A.A13.A.A.A2.A.A2.A.A$14.2A.A.A5.A.2A7.2A2.A
2.A3.A5.A.2A7.2pA2.pA2.pA3.pA5.pA.2pA7.2A2.A2.A3.A5.A.2A7.2A2.A2.A3.A
5.A.2A$14.A2.A.7A17.A.A.7A17.pA.pA.7pA17.A.A.7A17.A.A.7A$16.A10.2A5.
10A.A9.2A5.10pA.pA9.2pA5.10A.A9.2A5.10A.A8.3A5.2A$14.A.4A2.A.3A2.A.2A
7.A5.A.A.A.3A2.A.2pA7.pA5.pA.pA.pA.3pA2.pA.2A7.A5.A.A.A.3A2.A.2A7.A5.
A.A.A.3A2.A.2A$14.A7.2A4.2A.A.2A2.2A3.3A.A.A.2A4.2A.pA.2pA2.2pA3.3pA.
pA.pA.2pA4.2pA.A.2A2.2A3.3A.A.A.2A4.2A.A.2A2.2A3.3A.A.A.2A4.2A.A.3A$
12.2A.A.A.3A3.3A4.A2.A3.3A4.A2.A3.3A4.pA2.pA3.3pA4.pA2.pA3.3pA4.A2.A
3.3A4.A2.A3.3A4.A2.A3.3A4.A2.A3.3A4.A$8.A.A4.A.A.A3.2A3.3A.A.A.2A4.2A
.A.2A2.2A3.2ApA.pA.pA.2pA4.2pA.pA.2pA2.2pA3.2pAA.A.A.2A4.2A.A.2A2.2A
3.3A.A.A.2A4.2A.A.2A2.2A3.3A.A.A.A$8.A.A.2A.A2.A7.A5.A.A.A.3A2.A.2A7.
A5.pA.pA.pA.3pA2.pA.2pA7.pA5.A.A.A.3A2.A.2A7.A5.A.A.A.3A2.A.2A7.A5.A.
A.A$10.A4.2A.A.A6.3A.A9.2A5.10A.pA9.2pA5.10pA.A9.2A5.10A.A9.2A5.10A.A
5.2A.2A$9.A.3A4.A.A8.A.A.7A17.A.pA.7pA17.pA.A.7A17.A.A.7A17.A.A.4A4.A
$9.2A3.2A.A11.A3.A5.A.2A7.2A2.A2.A3.pA5.pA.2pA7.2pA2.pA2.pA3.A5.A.2A
7.2A2.A2.A3.A5.A.2A7.2A2.A2.A3.A4.2A$12.A14.A.A.A2.A.A2.A.A13.A.A.pA
2.pA.pA2.pA.pA13.pA.pA.A2.A.A2.A.A13.A.A.A2.A.A2.A.A12.A4.A3.2A.A.4A$
2A2.A4.2A.A.3A10.A.A.2A.A.A6.A10.2A.A.2pA.pA.pA6.pA10.2pA.pA.2A.A.A6.
A10.2A.A.2A.A.A6.A16.3A3.A4.A$A2.A2.A.A.A.A.A19.A5.3A19.pA5.3pA19.A5.
3A19.A5.3A20.2A.A.2A$.A.A.2A.A2.A2.A19.A4.A22.pA4.pA22.A4.A22.A4.A20.
2A2.A3.A.4A$3.A5.A.A.2A.2A23.A.A25.pA.pA25.A.A25.A.A20.A3.A$2.A2.3A.A
.A2.A27.2A26.2pA26.2A26.2A21.2A.A.2A$7.A.A2.A.A132.A.A2.A$7.A2.A.A.A.
A45.3V.3V78.A2.A.A$4.2A.A.2A3.A.A42.V4.V.V81.A.A.3A2.A$7.A3.A2.A43.3V
3.V.3V77.A.A.A5.A$2.4A.A.2A.A.A44.V4.V.V.V72.2A2.A2.A2.A.2A.A.A$6.2A
4.A51.V.3V76.A.A.A.A.A2.A2.A$4.A4.3A.A127.5A.A.2A4.A2.2A$4.2A.2A3.2A
133.Q$10.A130.2Q.2Q3.2Q$4.5A.A.2A4.A2.2A118.Q4.3Q.Q$8.A.A.A.A.A2.A2.A
120.2Q4.Q$4.2A2.A2.A2.A.2A.A.A117.4Q.Q.2Q.Q.Q$4.A.A.2A.A.A5.A124.Q3.Q
2.Q$8.A2.A.A.3A2.A120.2Q.Q.2Q3.Q.Q$8.A.A2.A.A128.Q2.Q.Q.Q.Q$6.A.A.A.A
2.A128.Q.Q2.Q.Q$6.A.A3.2A.A.2A120.Q2.3Q.Q.Q2.Q$8.A2.A3.A124.Q5.Q.Q.2Q
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9.A.3A4.A118.2Q2.Q4.2Q.Q.5Q$9.2A3.2A.2A130.Q$12.A133.2Q3.2Q.2Q$2A2.A
4.2A.A.5A127.Q.3Q4.Q$A2.A2.A.A.A.A.A132.Q4.2Q$.A.A.2A.A2.A2.A2.2A126.
Q.Q.2Q.Q.4Q$3.A5.A.A.2A.A.A126.Q2.Q3.Q$2.A2.3A.A.A2.A128.Q.Q3.2Q.Q.2Q
$7.A.A2.A.A128.Q.Q.Q.Q2.Q$7.A2.A.A.A.A128.Q.Q2.Q.Q$4.2A.A.2A3.A.A99.
2Q27.Q2.Q.Q.3Q2.Q$7.A3.A2.A101.Q.Q23.2Q.2Q.Q.Q5.Q$2.4A.A.2A.A.A105.Q
4.Q19.Q2.Q2.Q.2Q.Q.Q$6.2A4.A104.3Q5.Q19.Q.Q.Q.Q.Q2.Q2.Q$4.A4.3A.A102.
Q6.Q.Q.2Q.Q.Q10.3Q.Q.2Q4.Q2.2Q$4.2A.2A3.2A104.Q.Q2.Q.Q2.Q.Q.Q14.Q$10.
A106.2Q.Q5.Q3.Q11.Q.2Q3.2Q$4.5A.A.2A4.A2.2A97.7Q.Q.Q8.Q.Q4.3Q.Q$8.A.A
.A.A.A2.A2.A87.2Q5.2Q9.Q.3Q6.Q.Q.2Q4.Q$4.2A2.A2.A2.A.2A.A.A91.2Q.Q2.
3Q.Q.Q.Q5.Q7.Q2.Q.2Q.Q.Q$4.A.A.2A.A.A5.A90.3Q.Q.2Q4.2Q.Q.Q.3Q3.2Q3.Q.
Q.Q4.Q.Q$8.A2.A.A.3A2.A92.Q4.3Q3.Q2.Q4.3Q3.3Q.Q.Q.2Q$8.A.A2.A.A93.Q.Q
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26.2A!

If four signals are used instead of one, then the period shifts will be 19 and 109, which are coprime. Therefore, it's possible to get any period not less than 7.
Oscillators with periods 1-6:

Code: Select all

x = 110, y = 11, rule = B368/S012378
2o18b2o18b2o2bobo19b2o15b2o17bob2obo$20bo22b2obo18bobo13bo4b2o14bobob
2o$23bo19bo16b2o3bo16b4o2bo14b3o$22b2o18bob2o14bo2b2obo13bobo4bobo14bo
$41b2obo16b2o3bo13bobo2bobo15b2o$44bo18bo3b2o13bobob2o14bobo2bo$41b2o
20bob2o2bo11bo3bo16bobo2bo$64bo3b2o11bob3o14bobo4bobo$62bobo17bo17bobo
b2obobo$62b2o19bo18bo4bo$102bo4bo!

So rules B36/S123 to B368/S012378 are omniperiodic.
Here is a list of omniperiodic rules:

Code: Select all

B3/S23
B3/S238
B38/S23
B3(8)/S(0)234(78)
B3(8)/S(0)123(678)
B36(8)/S(0)123(78)
B4(678)/S(012)35678
B(45)7/S(01)234568
B2(5)7/S(01)234568
B3(5)7/S(01)234568
B2(78)/S(0)12(4)5(678)
B(24)5(68)/S(01)2345(7)8
B3/S(01)24567(8)

Total: 427 rules.
P.S. Rule B3/S23 will be included in this list after discovery of a p41 oscillator.
Edit (22 July 2023): p41 oscillator is known, so B3/S23 (Life), B38/S23 (Pedestrian Life) and B3/S238 (Eight Life) are omniperiodic.

[atavoidirc]

Edit (22 July 2023): p41 oscillator is known, so B3/S23 (Life), B38/S23 (Pedestrian Life) and B3/S238 (Eight Life) are omniperiodic.

What about B38/S238 (Honey Life)

[May13]

Edit (22 July 2023): p41 oscillator is known, so B3/S23 (Life), B38/S23 (Pedestrian Life) and B3/S238 (Eight Life) are omniperiodic.

What about B38/S238 (Honey Life)

There are currently no known p29 and p34 oscillators:

Only P29 and P34 oscillators are needed in order to prove omniperiodicity, the other oscillators on the following figure are B3[8]/S23[8]-compatible polyglots:

Code: Select all

x = 384, y = 335, rule = B38/S238History
18.2A28.2A10.2A.2A54.A$18.2A28.2A10.2A.2A23.2A29.A33.2A25.3A28.2A26.
2A31.2A25.A29.2A34.2A$88.2A29.A29.A.4A25.3A27.4A25.2A32.A24.A.A29.A
34.A.A$14.2A2.2A24.2A2.2A7.2A7.2A79.3A2.A.A23.3A2.2A24.A4.A24.2A31.A
25.2A3.2A25.A.2A28.A3.3A.A$14.2A2.2A24.2A2.2A7.2A7.2A115.2A23.2A4.2A
23.A31.A.3A24.A2.2A24.2A.A.A27.A6.2A$149.2A31.3A23.2A4.2A22.A.A30.A.A
2.A24.A.A31.A26.A3.A$18.2A28.2A7.2A7.2A82.A31.A26.A4.A24.A.A.3A22.2A.
A3.A.A22.2A2.A29.2A22.2A6.A$18.2A8.A.2A.2A.A11.2A7.2A7.2A114.A27.4A
26.A.4A22.2A.A.A2.A23.2A3.2A21.2A2.A3.2A20.A.3A3.A$28.2A.A.A.2A174.2A
28.A30.4A28.A.A21.A2.A.2A.A24.A.A$18.2A28.2A7.2A7.2A237.A24.2A.A2.A
25.2A$18.2A28.2A7.2A7.2A204.2A60.2A$272.2A$18.2A28.2A7.2A7.2A$18.2A
28.2A7.2A7.2A2$18.2A28.2A10.2A.2A$18.2A28.2A10.2A.2A24$8.2A8.2A28.2A.
A8.2A.2A88.A$8.2A8.2A28.A.2A8.2A.2A25.2A.2A24.A33.A25.A5.A29.2A22.2A
10.2A20.A27.2A5.A25.2A25.2A$91.A.A.A23.3A7.2A23.A23.A.A3.A.A28.A23.2A
10.2A19.A.A27.A4.2A21.2A.A2.A2.2A19.A2.A$4.2A2.2A4.2A2.2A25.2A6.2A2.
2A7.2A23.A4.A25.A6.A24.2A24.2A.2A31.A22.2A10.2A19.A.A3.2A20.A29.A3.A.
A3.A18.A.A.A$4.2A2.2A4.2A2.2A25.2A6.2A2.2A7.2A20.2A.2A4.A23.2A4.A.A
23.3A23.2A3.2A25.2A2.2A22.A12.A16.2A.A.2A3.A20.2A3.3A2.A.A19.2A.A.3A
18.3A.A$88.2A.A6.A28.2A25.2A24.2A.2A25.A2.A5.A18.A.A10.A.A16.A.A6.A.
2A24.A.4A15.4A.A.A.A20.3A$8.2A8.2A33.2A2.2A7.2A23.A.2A4.A23.2A83.A2.A
4.4A2.A15.A.2A8.2A.A14.A2.A.2A.2A.A2.A15.5A8.2A15.A2.A.A.2A$8.2A8.2A
8.A.2A.2A.A17.A2.2A7.2A23.A.2A3.2A23.2A31.A51.2A.A.A2.A3.3A45.2A.A.A.
A2.A.A17.A2.A.A28.A2.A26.A$28.2A.A.A.2A12.2A.A39.A3.A26.2A27.A4.A53.A
2.2A3.A51.A.2A.2A2.2A17.A.A.3A27.A$8.2A8.2A29.2A.2A3.2A7.2A25.7A20.2A
26.2A3.2A4.3A49.2A6.2A18.2A5.A4.2A18.A.A3.A.A18.2A.2A3.A54.A2.A$8.2A
8.2A37.2A7.2A31.A19.A.A4.2A20.2A3.A.A3.4A61.2A13.2A5.3A2.2A19.2A.A.A.
A20.A3.2A53.3A2.3A$45.2A48.2A22.A6.A27.2A4.A2.2A52.A5.A.A20.3A25.A.A.
A21.A.2A33.A.3A18.2A2.2A$8.2A8.2A25.2A10.2A7.2A27.2A21.2A7.3A26.A59.A
.A4.A50.A.A24.A.A33.A.3A19.A2.A$8.2A8.2A37.2A7.2A61.A85.2A.2A2.2A51.
2A60.A21.A.A2.A.A$216.A2.A23.3A97.2A14.2A4.2A$8.2A8.2A25.2A.2A.2A.2A
4.2A.2A89.2A56.2A2.2A.2A18.2A2.3A5.2A90.A.A$8.2A8.2A25.2A.2A.2A.2A4.
2A.2A89.2A56.A4.A.A19.2A4.A5.2A77.2A13.A$210.A.A5.A112.A13.2A$210.2A
119.A.A$215.2A6.2A13.A.2A8.2A.A78.2A$216.A3.2A2.A13.A.A10.A.A85.A$
213.3A3.A2.A.A.2A11.A12.A82.3A.A$213.A2.4A4.A2.A11.2A10.2A82.3A.A$
216.A5.A2.A13.2A10.2A$219.2A2.2A14.2A10.2A$219.A123.A$220.A120.A2.A$
219.2A119.2A.A.A2.A$339.A.A.A.4A$337.3A.A.2A$336.A3.A.A3.A$336.2A2.A
2.A.2A$341.2A3$333.14F$346.F$346.F3$5.2A.A9.2A28.2A.A8.2A.2A87.A32.2A
21.A37.A6.A15.2A.2A27.2A32.2C30.A$5.A.2A9.2A28.A.2A8.2A.2A28.2A26.2A
29.3A4.2A24.2A21.3A33.A.A4.3A16.A.A.A.2A22.A2.A32.C28.A.A$93.2A25.2A
33.A3.2A50.A33.A.A2.A18.A2.A.A.A21.A2.A.A30.C29.A.A11.2A$2.2A6.2A2.2A
2.2A25.2A6.2A2.2A7.2A52.2A3.2A3.3A21.2A54.A27.2A5.A4.2A16.A.2A2.A.A.A
23.A31.5C19.2A3.A2.A11.2A$2.2A6.2A2.2A2.2A25.2A6.2A2.2A7.2A21.A30.3A
2.2A3.4A76.A2.A25.A28.A9.3A.A15.A.2A36.C19.2A4.A.A$88.A.A3.A27.A.3A2.
2A2.A32.2A21.A49.A.A27.3A9.2A16.A32.4C29.A.A$10.2A6.2A37.2A7.2A21.A4.
A55.2A14.A17.A4.2A23.2A24.2A5.A23.A.A.A2.2A3.2A47.C2.C31.A$11.A6.2A8.
A.2A.2A.A13.2A5.2A7.2A25.A35.A20.A13.A.A11.2A3.A.A5.A23.A30.A.2A3.2A
18.A.A.A2.2A3.A$6.2A.A18.2A.A.A.2A13.A2.A43.2A21.3A6.3A20.A11.2A12.2A
2.2A.A5.2A20.A.A31.2A4.2A17.2A.A2.2A2.2A.A17.2A41.C$6.2A.2A7.2A32.2A
3.2A7.2A25.2A2.2A21.3A6.3A19.2A6.A23.2A27.2A57.A3.A.A2.A.A.A13.2A2.2A
.A5.2A18.C7.3C3.C.C$18.2A37.2A7.2A25.2A2.2A23.A27.A6.3A62.2A48.3A2.A
2.A6.A10.2A3.A.A5.2A17.C.C6.C.C4.C$3.2A88.2A54.A.2A4.A.A36.2A22.A.A
52.2A3.A2.5A17.A24.C.C6.3C$3.2A13.2A25.2A6.2A2.2A7.2A30.A19.A2.2A2.3A
.A18.A2.A.A5.A29.2A5.2A22.A53.A2.4A.A24.A23.C$18.2A25.2A6.2A2.2A7.2A
29.A4.A15.4A3.2A2.3A16.2A2.A35.A2.A28.2A51.A.2A3.A.A2.3A$97.A3.A.A15.
3A3.2A3.2A56.3A83.A2.3A2.A4.A19.A23.C$2.2A.2A.2A.2A5.2A28.2A.A8.2A.2A
37.A27.2A31.2A24.A32.A2.A49.2A5.A.2A19.A24.C.C6.3C$2.2A.2A.2A.2A5.2A
28.A.2A8.2A.2A64.2A25.2A5.A.A59.A51.3A2.A.A13.2A3.A.A5.2A17.C.C6.C.C
4.C$97.2A58.A7.A58.A52.A2.A.A.A13.2A2.2A.A5.2A18.C7.3C3.C.C$97.2A55.
3A8.2A22.2A34.3A13.2A4.2A29.5A.2A17.2A41.C$154.A34.2A36.A13.2A3.2A.A
33.A2.A$248.A5.2A21.A.4A2.2A45.C2.C$254.A.A20.2A.A51.4C$256.A81.C$
244.2A4.A5.2A76.5C$245.A2.A.A83.C$242.3A4.A.A84.C$242.A6.A85.2C5$346.
F$346.F$333.14F2$187.14F$200.F$200.F$200.F$200.F$5.2A.A9.2A24.2A7.2A
5.2A.2A24.2A7.2A28.2A23.A61.2A22.A29.A12.A28.A21.2A4.3A17.2A20.2A$5.A
.2A9.2A24.2A7.2A5.2A.2A24.2A7.2A28.A24.A39.C21.2A22.3A27.3A8.3A27.A2.
A18.A25.A2.A19.2A$122.2A.2A.A4.2A18.A37.3C14.2A32.A29.A6.A30.A2.A2.A
18.A5.A16.A$2.2A6.2A2.2A2.2A24.2A7.2A2.2A7.2A52.A2.A.A.A5.A18.2A27.2C
7.C17.A.A30.2A28.2A6.2A31.A3.A14.A3.A5.2A15.A$2.2A6.2A2.2A2.2A24.2A7.
2A2.2A7.2A52.2A4.A8.A14.A31.C7.2C17.A106.A11.4A3.A4.A17.A.2A20.2A$
134.2A13.A4.A3.2A22.C.C30.A117.A9.A16.2A18.A.2A$18.2A24.2A7.2A2.2A7.
2A21.A9.A49.A7.A.A23.2C29.2A123.A3.A36.A$7.2A9.2A8.A.2A.2A.A7.A2.A.2A
2.2A2.2A7.2A20.A.A7.A.A33.2A15.3A3.A57.A82.A45.2A34.A$7.A2.A17.2A.A.A
.2A9.2A.2A37.A2.A5.A2.A17.A10.A3.A.A20.2A20.2C33.2A5.2A68.2A6.3A34.2A
2.3A5.A11.2A13.2A4.A2.A$9.2A7.2A33.2A2.2A7.2A21.2A7.2A18.3A7.3A3.A44.
C40.A.A51.2A14.A10.A14.A18.2A5.A4.2A.A8.2A12.A2.A4.2A$18.2A33.2A2.2A
7.2A20.A11.A20.A6.A.A4.3A41.C.C40.A30.2A18.2A.A11.A.A9.2A14.2A24.A3.A
3.A22.A2.A$120.A.A14.A42.2C40.2A28.A2.A17.2A.A11.2A25.2A25.A.A5.A21.
2A.2A$2.2A6.2A6.2A33.2A2.2A7.2A52.2A31.A33.3C13.2C4.2A39.A2.A.A19.A
97.2A$2.2A6.2A6.2A33.2A2.2A7.2A20.A11.A52.2A31.C3.C12.C6.A32.3A4.A3.A
51.2A$89.2A7.2A20.2A26.2A.A.A.3A28.2C.2C10.C.C6.A.A30.A2.A3.A33.A20.
2A14.2A18.2A22.A10.A$5.2A.A9.2A33.2A5.2A.2A23.A2.A5.A2.A19.A8.A4.2A
12.A.2A.A.A.A43.2C8.2A5.2A23.A.2A6.A16.2A11.A.2A19.A14.A19.2A21.A.A8.
A.A$5.A.2A9.2A33.2A5.2A.2A23.A.A7.A.A21.A5.A.A.A2.A18.A2.3A57.A33.2A
16.A.A11.A.2A35.3A40.A4.2A4.A$89.A9.A21.2A4.A.2A.2A21.2A3.A19.2C35.2A
50.A14.2A38.A43.A4.A$127.A29.3A19.C.C10.2C.2C20.A50.2A98.A4.A$126.2A
29.A21.C12.C3.C26.A144.A4.A$178.2C13.3C26.A.A81.A62.A2.A$201.2C20.2A
81.A3.A56.A.A2.A.A$89.2A7.2A101.C.C12.2A23.2A63.A2.A2.A54.2A4.2A$89.
2A7.2A103.C12.2A22.A6.2A.A28.2A6.2A20.A2.A$203.2C38.A3.A2.A29.A6.A23.
A$239.A3.A4.3A26.3A8.3A$198.2C38.A.A2.A33.A12.A$198.C.C37.A2.A$191.2C
7.C38.2A$192.C7.2C$189.3C$189.C4$251.2A$182.F68.A$182.F69.3A$182.F71.
A$182.F$2A7.2A7.2A102.A59.16F$2A7.2A7.2A100.3A$119.A$2A7.2A3.2A2.2A
84.2A13.2A$2A7.2A3.2A2.2A85.A$105.A.A$2A7.2A7.2A86.2A$A2.A.2A2.2A7.2A
96.A$2.2A.2A108.A$9.2A7.2A97.A$9.2A7.2A95.3A2$9.2A7.2A$9.2A7.2A$128.A
$9.2A7.2A73.2A24.2A5.A.A$9.2A7.2A74.A24.A7.2A$92.A13.2A12.3A$90.4A15.
2A11.A$89.A14.2A.A$88.A2.3A10.2A3.A.2A$89.2A2.A10.2A.A3.2A$91.2A16.A.
2A$91.A14.2A$92.A6.3A7.2A26.A$89.3A7.A38.2A$89.A5.2A3.A11.A24.2A$96.A
14.A.A$93.3A15.2A$93.A2$155.A$136.2A15.3A$135.A.A14.A$110.2A24.A11.A
3.2A5.A$109.2A38.A7.3A$111.A26.2A7.3A6.A$141.2A14.A$136.2A.A16.2A$
136.2A3.A.2A10.A2.2A$136.2A.A3.2A10.3A2.A$141.A.2A14.A$126.A11.2A15.
4A$126.3A12.2A13.A$120.2A7.A24.A$120.A.A5.2A24.2A$120.A4$131.3A$131.A
$133.A$132.A$141.2A$141.A.A$143.A$128.2A13.2A$129.A$126.3A$126.A10$2A
7.2A7.2A.A88.2A$2A7.2A7.A.2A87.A2.A$98.A13.A$2A7.2A4.2A6.2A72.A.A12.A
$2A7.2A4.2A6.2A73.A7.A3.A.A$105.A3.A.A$2A7.2A12.2A71.5A4.A4.A$A2.A.2A
2.2A13.A70.A4.A5.4A4.2A$2.2A.2A12.2A.A71.A.2A16.2A2.2A$9.2A8.2A.2A67.
A2.A.A15.2A.A2.2A$9.2A79.A.A.A9.A7.3A$16.2A73.A2.A4.3A3.2A$9.2A5.2A
76.2A3.2A3.2A$9.2A88.A13.2A$113.2A$9.2A4.2A.2A.2A.2A$9.2A4.2A.2A.2A.
2A72.A.A$93.2A4.2A$92.A2.A3.A7.2A$95.A10.A2.A2.2A$95.A10.A.A4.A2.A$
89.A3.A.A11.A5.A.A.A$88.A3.A.A15.2A.A2.A$88.A4.A16.A2.A$89.4A4.2A8.A
4.A$97.2A2.2A4.5A$95.2A.A2.2A$95.3A11.A$108.A.A$109.A$96.2A$96.2A9$2A
7.2A7.2A.A93.2A$2A7.2A7.A.2A93.A.A$117.A4.2A$2A7.2A4.2A6.2A88.4A.2A2.
A2.A$2A7.2A4.2A6.2A88.A2.A3.A.A.2A$116.A.A.A.A$2A7.2A106.2A.A.A$A2.A.
2A2.2A9.2A6.A.2A.2A.A84.A$2.2A.2A13.A2.A4.2A.A.A.2A$9.2A11.2A83.2A$9.
2A97.A8.A$108.A.A5.2A$9.2A4.2A6.2A84.2A$9.2A4.2A6.2A98.A$124.A$9.2A7.
2A.A100.3A$9.2A7.A.2A$113.A$113.2A$112.A.A4.2A22.A$119.A21.3A$120.3A
17.A$122.A17.2A2$133.A$134.2A12.2A$133.2A14.A$91.2A56.A.2A$92.A9.2A
37.A5.3A2.A$90.A10.A.A37.2A3.A3.2A$90.5A8.A5.2A35.2A.A$95.A13.A22.2A
15.A$92.3A12.A.A21.A.A12.3A$91.A15.2A22.A13.A$91.A.2A35.2A5.A8.5A$89.
2A3.A3.2A37.A.A10.A$88.A2.3A5.A37.2A9.A$88.2A.A56.2A$91.A14.2A$91.2A
12.2A$107.A2$99.2A17.A$100.A17.3A$97.3A21.A$97.A22.2A4.A.A$126.2A$
127.A2$116.3A$116.A$117.A$130.2A$123.2A5.A.A$123.A8.A$132.2A2$119.A$
118.A.A.2A$118.A.A.A.A$115.2A.A.A3.A2.A$115.A2.A2.2A.4A$117.2A4.A$
123.A.A$124.2A!

Edit: I didn't noticed a p34 because Catagolue shows:

Code: Select all

This evolutionary sequence works in multiple rules, from b3s23 through to b34e7c8s234ce5r6-ac.

Indeed, two cells have a B8 configuration, but their next state does not affect the oscillator.

[yujh]

These is a p34 in honeylife:

Code: Select all

x = 28, y = 39, rule = B38/S238
19bo$18bobo$19bo2$17b5o$4b2o6b2o3bo4bo$4bo7bo7bo2bo$5b3ob2obo7b2obo2bo
$2o5bobobo5bo5bobobo$bo8bo5bobo4bo2bo$bobo12bo2bo2b2o$2b2o13b2o3$9bobo
$9bobo$9b3o6$16b3o$16bobo$16bobo3$9b2o13b2o$4b2o2bo2bo12bobo$bo2bo4bob
o5bo8bo$obobo5bo5bobobo5b2o$bo2bob2o7bob2ob3o$4bo2bo7bo7bo$5bo4bo3b2o
6b2o$6b5o2$8bo$7bobo$8bo!

However no p29 are known yet.

[May13]

I found a topic "Omniperiodicity based on XOR replicators". This post shows that "Vote 4/5" (B4678/S35678) is omniperiodic. XOR oscillators works in rules B4(678)/S(012)35678. For example, here's a p17 oscillator found using w150finder.py (shown in minrule and maxrule):

Code: Select all

x = 513, y = 513, rule = B4/S35678
2b2ob2ob2ob2ob2ob4o2b4o2b4o4bo2b2o2bo4bo2bo3b2o3bo2bo2bo3b4o3bo2b2o3bo
b2obo3b2ob2o2bob2obo2b2ob3o3b4o3b3o3b3o4b3o3bo2b2obo2bob2o2bo2b3obo2bo
b3o2b2ob3o4b3ob2ob2ob3o2b3ob2ob4ob2o2b2ob4o4b2o4b2o4bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo4b2o4b2o4b4ob2o2b2ob4ob2ob3o2b3ob2ob2ob3o4b3ob2o2b3obo
2bob3o2bo2b2obo2bob2o2bo3b3o4b3o3b3o3b4o3b3ob2o2bob2obo2b2ob2o3bob2obo
3b2o2bo3b4o3bo2bo2bo3b2o3bo2bo4bo2b2o2bo4b4o2b4o2b4ob2ob2ob2ob2ob2o$b
511o$513o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b
2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b
3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o
$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b
2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o
507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o
$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b
3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b
2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b
3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$
3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b
3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$
b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b
2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b
2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$
3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o
507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b
2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b
2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o
507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o
507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b
3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b
2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$
3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b
2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o
$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b
2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$
3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b
2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$
b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b
2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o
507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b
3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b
2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$
513o$b511o$2b2ob2ob2ob2ob2ob4o2b4o2b4o4bo2b2o2bo4bo2bo3b2o3bo2bo2bo3b
4o3bo2b2o3bob2obo3b2ob2o2bob2obo2b2ob3o3b4o3b3o3b3o4b3o3bo2b2obo2bob2o
2bo2b3obo2bob3o2b2ob3o4b3ob2ob2ob3o2b3ob2ob4ob2o2b2ob4o4b2o4b2o4bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo4b2o4b2o4b4ob2o2b2ob4ob2ob3o2b3ob2ob2ob3o4b
3ob2o2b3obo2bob3o2bo2b2obo2bob2o2bo3b3o4b3o3b3o3b4o3b3ob2o2bob2obo2b2o
b2o3bob2obo3b2o2bo3b4o3bo2bo2bo3b2o3bo2bo4bo2b2o2bo4b4o2b4o2b4ob2ob2ob
2ob2ob2o!

Code: Select all

x = 513, y = 513, rule = B4678/S01235678
2b2ob2ob2ob2ob2ob4o2b4o2b4o4bo2b2o2bo4bo2bo3b2o3bo2bo2bo3b4o3bo2b2o3bo
b2obo3b2ob2o2bob2obo2b2ob3o3b4o3b3o3b3o4b3o3bo2b2obo2bob2o2bo2b3obo2bo
b3o2b2ob3o4b3ob2ob2ob3o2b3ob2ob4ob2o2b2ob4o4b2o4b2o4bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo4b2o4b2o4b4ob2o2b2ob4ob2ob3o2b3ob2ob2ob3o4b3ob2o2b3obo
2bob3o2bo2b2obo2bob2o2bo3b3o4b3o3b3o3b4o3b3ob2o2bob2obo2b2ob2o3bob2obo
3b2o2bo3b4o3bo2bo2bo3b2o3bo2bo4bo2b2o2bo4b4o2b4o2b4ob2ob2ob2ob2ob2o$b
511o$513o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b
2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b
3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o
$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b
2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o
507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o
$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b
3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b
2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b
3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$
3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b
3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$
b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b
2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b
2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$
3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o
507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b
2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b
2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o
507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o
507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b
3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b
2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$
3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$b
2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o
507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o
$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b
3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b
2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o507b3o$3o507b
3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$
3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$3o
507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$3o507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b
2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$
b2o507b2o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$b
2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$b2o
507b2o$b2o507b2o$3o507b3o$b2o507b2o$b2o507b2o$b2o507b2o$b2o507b2o$3o
507b3o$3o507b3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b
3o$3o507b3o$3o507b3o$b2o507b2o$b2o507b2o$3o507b3o$3o507b3o$3o507b3o$3o
507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b
2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$3o507b3o$b2o507b2o$3o507b3o$
513o$b511o$2b2ob2ob2ob2ob2ob4o2b4o2b4o4bo2b2o2bo4bo2bo3b2o3bo2bo2bo3b
4o3bo2b2o3bob2obo3b2ob2o2bob2obo2b2ob3o3b4o3b3o3b3o4b3o3bo2b2obo2bob2o
2bo2b3obo2bob3o2b2ob3o4b3ob2ob2ob3o2b3ob2ob4ob2o2b2ob4o4b2o4b2o4bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo4b2o4b2o4b4ob2o2b2ob4ob2ob3o2b3ob2ob2ob3o4b
3ob2o2b3obo2bob3o2bo2b2obo2bob2o2bo3b3o4b3o3b3o3b4o3b3ob2o2bob2obo2b2o
b2o3bob2obo3b2o2bo3b4o3bo2bo2bo3b2o3bo2bo4bo2b2o2bo4b4o2b4o2b4ob2ob2ob
2ob2ob2o!

So rules B4(678)/S(012)35678 are omniperiodic.

[hotcrystal0]

[POST MOVED]

[hotdogPi]

hotcrystal0, that rule is not Life-like.

[May13]

I found another rule 150 emulator:

Code: Select all

x = 511, y = 3, rule = B7/S234568
511o$o2bo2bo2bo2bo2b2obo2bo2bo2bo2b2o4bo2bo2bo2b2o3b2o2bo2bo2b2o3b3obo
2bo2b2o3b3o4bo2b2o3b3o3b2o2b2o3b3o3b3ob2o3b3o3b3o2bo3b3o3b3o2b2o2b3o3b
3o2b2ob4o3b3o2b2ob2obo3b3o2b2ob2ob2o2b3o2b2ob2ob2ob4o2b2ob2ob2ob2obo2b
2ob2ob2ob2ob2ob2ob2ob2ob2ob2o2bob2ob2ob2ob2o2b4ob2ob2ob2o2b3o2b2ob2ob
2o2b3o3bob2ob2o2b3o3b4ob2o2b3o3b3o2b2o2b3o3b3o3bo2b3o3b3o3b2ob3o3b3o3b
2o2b2o3b3o3b2o2bo4b3o3b2o2bo2bob3o3b2o2bo2bo2b2o3b2o2bo2bo2bo4b2o2bo2b
o2bo2bob2o2bo2bo2bo2bo2bo$511o!

This emulator works in at least 32 rules:

Code: Select all

B(45)7/S(01)234568
B2(5)7/S(01)234568
B3(5)7/S(01)234568

From left to right: p1, p2, p7

Code: Select all

x = 147, y = 3, rule = B7/S234568
3o7b5o5b127o$obo7bo3bo5bo2bo2bob2o2bo4b2o2b2o3b2ob3o3bo2b3o2b2o2b4ob2o
2bob2ob2ob2ob2obo2b2ob4o2b2o2b3o2bo3b3ob2o3b2o2b2o4bo2b2obo2bo2bo$3o7b
5o5b127o!

Code: Select all

x = 173, y = 13, rule = B27/S234568
4bo18bo20bo123bo$3b3o16b3o18b127o$24bobo$bob3obo15b5o13bob127obo$2obob
ob2o31b2obo2bo2bob2o2bo4b2o2b2o3b2ob3o3bo2b3o2b2o2b4ob2o2bob2ob2ob2ob
2obo2b2ob4o2b2o2b3o2bo3b3ob2o3b2o2b2o4bo2b2obo2bo2bob2o$bob3obo13bob5o
bo11bob127obo$20b2obo3bob2o$3b3o15bob5obo13b127o$4bo39bo123bo$23b5o$
24bobo$22b3o$23bo!

Code: Select all

x = 186, y = 19, rule = B37/S234568
26bo$25bobo$7bo17b3o$6bobo43b2obo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bob2o$b2o3b3o16b3o24b2ob127ob2o$obo18b2obo3bob2o$b
2ob3o14b2ob5ob2o20b2ob127ob2o$4bobo44bobobo2bo2bob2o2bo4b2o2b2o3b2ob3o
3bo2b3o2b2o2b4ob2o2bob2ob2ob2ob2obo2b2ob4o2b2o2b3o2bo3b3ob2o3b2o2b2o4b
o2b2obo2bo2bobobo$4b3ob2o11b2ob5ob2o20b2ob127ob2o$8bobo9bobobo3bobobo$
2b3o3b2o11b2ob5ob2o20b2ob127ob2o$2bobo47b2obo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bob2o$3bo17b2ob5ob2o$21b2obo3bob2o$25b
3o2$25b3o$25bobo$26bo!

Edit: another rule 150 emulator, works in 128 rules (B2(78)/S(0)12(4)5(678)):

Code: Select all

x = 167, y = 12, rule = B2/S125
2bo8bo3bo2bo6bo5bo3bo125bo3bo$b2obo5b2obobob2obo2bob2o3b2obob2o123b2ob
ob2o$4b2o7bo7b2o10bo129bo$2o9bo3b5o4b2o5bo3b127o3bo$bob2o7bobo5bobobo
7bobo127bobo$3bo8bobo2bo2bobo9bobob2ob2obo2b2ob4o2b2o2b3o2bo3b3ob2o3b
2o2b2o4bo2b2obo2bo2bo2bo2bob2o2bo4b2o2b2o3b2ob3o3bo2b3o2b2o2b4ob2o2bob
2ob2obobo$12bobo2bo2bobo9bobob2ob2obo2b2ob4o2b2o2b3o2bo3b3ob2o3b2o2b2o
4bo2b2obo2bo2bo2bo2bob2o2bo4b2o2b2o3b2ob3o3bo2b3o2b2o2b4ob2o2bob2ob2ob
obo$12bobo5bobobo7bobo127bobo$11bo3b5o4b2o5bo3b127o3bo$13bo7b2o10bo
129bo$10b2obobob2obo2bob2o3b2obob2o123b2obob2o$11bo3bo2bo6bo5bo3bo125b
o3bo!

Edit 2: another rule 150 emulator, works in 128 rules (B(24)5(68)/S(01)2345(7)8):

Code: Select all

x = 277, y = 44, rule = B5/S23458
bo35bo2bo23bo2bo49bo2bo145bo2bo$3o33b3ob2o21b2ob3o47b3ob2o143b2ob3o$bo
35bo29bo49bo151bo$40bo23bo55bo145bo$36b2ob3o21b3ob2o47b2ob3o143b3ob2o$
35b2o2bob2o19b2obo2b2o45b2o2bob2o141b2obo2b2o$31bo2b2o4b2o4bo11bo4b2o
4b2o2bo37bo2b2o4b2o4bo133bo4b2o4b2o2bo$30b3obo5bo4b3o9b3o4bo5bob3o35b
3obo5bo4b3o131b3o4bo5bob3o$31bo7bo4b2ob2o7b2ob2o4bo7bo37bo7bo4b2ob2o
129b2ob2o4bo7bo$34b2o2b2o5b2o2b7o2b2o5b2o2b2o43b2o2b2o5b2o2b129o2b2o5b
2o2b2o$30b2ob2ob2o2bo23bo2b2ob2ob2o35b2ob2ob2o2bo145bo2b2ob2ob2o$31bo
2b3o4b4o2b2o7b2o2b4o4b3o2bo37bo2b3o4b4o2b2o129b2o2b4o4b3o2bo$35bo5bobo
4b2o5b2o4bobo5bo45bo5bobo4b2o127b2o4bobo5bo$41b2o19b2o57b2o141b2o$38bo
2bo2b4o2b5o2b4o2bo2bo51bo2bo2b4o2b127o2b4o2bo2bo$37b3o4bobo4b3o4bobo4b
3o49b3o4bobo4b125o4bobo4b3o$36b2obo4b2o13b2o4bob2o47b2obo4b2o135b2o4bo
b2o$37b2o2bo2bo2b4o3b4o2bo2bo2b2o49b2o2bo2bo2b4o125b4o2bo2bo2b2o$38bo
2b2o4bobo5bobo4b2o2bo51bo2b2o4bobo127bobo4b2o2bo$39bo2bo4b2o7b2o4bo2bo
53bo2bo4b2o129b2o4bo2bo$39bo4bo2bo2b5o2bo2bo4bo53bo4bo2bo2b127o2bo2bo
4bo$39bo4b2o4bo3bo4b2o4bo53bo4b2o4bo2bo2bob2o2bo4b2o2b2o3b2ob3o3bo2b3o
2b2o2b4ob2o2bob2ob2ob2ob2obo2b2ob4o2b2o2b3o2bo3b3ob2o3b2o2b2o4bo2b2obo
2bo2bo4b2o4bo$39bo4b2o4bo3bo4b2o4bo53bo4b2o4bo2bo2bob2o2bo4b2o2b2o3b2o
b3o3bo2b3o2b2o2b4ob2o2bob2ob2ob2ob2obo2b2ob4o2b2o2b3o2bo3b3ob2o3b2o2b
2o4bo2b2obo2bo2bo4b2o4bo$39bo4bo2bo2b5o2bo2bo4bo53bo4bo2bo2b127o2bo2bo
4bo$39bo2bo4b2o7b2o4bo2bo53bo2bo4b2o129b2o4bo2bo$38bo2b2o4bobo5bobo4b
2o2bo51bo2b2o4bobo127bobo4b2o2bo$37b2o2bo2bo2b4o3b4o2bo2bo2b2o49b2o2bo
2bo2b4o125b4o2bo2bo2b2o$36b2obo4b2o13b2o4bob2o47b2obo4b2o135b2o4bob2o$
37b3o4bobo4b3o4bobo4b3o49b3o4bobo4b125o4bobo4b3o$38bo2bo2b4o2b5o2b4o2b
o2bo51bo2bo2b4o2b127o2b4o2bo2bo$41b2o19b2o57b2o141b2o$35bo5bobo4b2o5b
2o4bobo5bo45bo5bobo4b2o127b2o4bobo5bo$31bo2b3o4b4o2b2o7b2o2b4o4b3o2bo
37bo2b3o4b4o2b2o129b2o2b4o4b3o2bo$30b2ob2ob2o2bo23bo2b2ob2ob2o35b2ob2o
b2o2bo145bo2b2ob2ob2o$34b2o2b2o5b2o2b7o2b2o5b2o2b2o43b2o2b2o5b2o2b129o
2b2o5b2o2b2o$31bo7bo4b2ob2o7b2ob2o4bo7bo37bo7bo4b2ob2o129b2ob2o4bo7bo$
30b3obo5bo4b3o9b3o4bo5bob3o35b3obo5bo4b3o131b3o4bo5bob3o$31bo2b2o4b2o
4bo11bo4b2o4b2o2bo37bo2b2o4b2o4bo133bo4b2o4b2o2bo$35b2o2bob2o19b2obo2b
2o45b2o2bob2o141b2obo2b2o$36b2ob3o21b3ob2o47b2ob3o143b3ob2o$40bo23bo
55bo145bo$37bo29bo49bo151bo$36b3ob2o21b2ob3o47b3ob2o143b2ob3o$37bo2bo
23bo2bo49bo2bo145bo2bo!

[Layz Boi]

W150 emulator in B3/S(01)24567(8):

Code: Select all

x = 26, y = 28, rule = B3/S24567
3bob2ob2ob2ob2ob2ob2obo$3b21o$23b3o$b21obobo$2b2ob2ob2ob2ob2ob2ob4o$ob
2ob2o11b2ob3o$3o19bo$2b2ob2o11b2ob3o$2b2ob2ob2ob2ob2ob2ob4o$b21obobo$
23b3o$3b21o$3bob2ob2ob2ob2ob2ob2obo3$3bob2ob2ob2ob2ob2obo$3b18o2$b22o$
2b2ob2ob2ob2ob2ob2ob2o$ob2o16b2obo$3o18b3o$2b2o16b2o$2b2ob2ob2ob2ob2ob
2ob2o$b22o2$3b18o$3bob2ob2ob2ob2ob2obo!

[confocaloid]

Is there a collection of omniperiodic CA?

I think this thread is linked from omniperiodic and might be the answer for Life-like CA. For other rulespaces, I don't know what is the exact status, but the short discussion viewtopic.php?p=164539#p164539 is relevant for multistate cellular automata.
(edit: clarification)

[WhiteHawk]

I have recently been investigating the Snark rulespace (B3/S23 to B34cq8/S234c5e6n8) and Snark 64 rulespace (B3/S23 to B34c8/S234cz5e8). The only rules I could find which have corresponding Omniperiodicity proofs are B3/S23 (Life), B34c/S23 (4diagonal), B38/S23 (Pedestrian Life), B34c8/S23, B3/S234c (Conway++), B34c/S234c (Honey Ring provides the missing p17), B38/S234c, B3/S238, B34c/S234c8, B3/S236n, and B3/S234c6n.

Both of these rulespaces already have a finite set of rules left to prove, though there definitely are some hurdles which need to be tackled first (e.g. there is no P13 which covers all of Snark64's rulespace). I am still working on tabulating all known oscillator periods, Life or Alien, for the 160 total rules covered by both rulespaces. Expect further updates soon.

[confocaloid]

I have recently been investigating the Snark rulespace (B3/S23 to B34cq8/S234c5e6n8) and Snark 64 rulespace (B3/S23 to B34c8/S234cz5e8). [...]

Are you accounting for different catalyst variants in Snark/Snark64? Changing catalysts may affect the minimal set of rules that are required for the glider-reflecting reaction to proceed in the same way.

As long as people are investigating/covering entire rulespaces like this (and are not posting one CA at a time), maybe it would make sense to extend the scope of this forum thread, from the set of 2^18 Life-like CA, to arbitrary range-1 two-state cellular automata on square/hexagonal/triangular grid.

[WhiteHawk]

I have recently been investigating the Snark rulespace (B3/S23 to B34cq8/S234c5e6n8) and Snark 64 rulespace (B3/S23 to B34c8/S234cz5e8). [...]

Are you accounting for different catalyst variants in Snark/Snark64? Changing catalysts may affect the minimal set of rules that are required for the glider-reflecting reaction to proceed in the same way.

As long as people are investigating/covering entire rulespaces like this (and are not posting one CA at a time), maybe it would make sense to extend the scope of this forum thread, from the set of 2^18 Life-like CA, to arbitrary range-1 two-state cellular automata on square/hexagonal/triangular grid.

I am simply going off of the rulespace as defined on the respective pages of Snark and Snark64 on the lifewiki (I think there isn't much hope for slight variations that don't affect how snarks work, but I obviously could be wrong). This can be found right in the main description box

What I think needs to be done in this instance is splitting Snark/Snark64 rules investigation off into it's own thread (some of the 160 rules which are part of both rulespaces are very far off from omniperiodicity, [Snarkperiodicity is my made-up term for using stable reflectors/Snarks to solve omniperiodicity]).

EDIT: it seems snark works under B4z, but I can't figure out where it differs.

DOUBLE EDIT: Ah. 31.4 recovers 1 transition faster under B4z. It should also be noted that B4z kills Caterer, among other things, which could potentially have ramifications for LCM solutions to periods.

[hotdogPi]

Onto your other point, I do agree that what constitutes a "Life-like" cellular automota is a very vague term (how would a "Life-unlike" rule differ from a so-called "life-like" rule, and where/how far away from life would the distinction be), but I most certainly am not going to go looking into multistate rules/hexagonal rules for omniperiodicity proof.

There is a very clear-cut definition for "Life-like". It's a synonym of outer-totalistic.

As soon as you start adding letters in the rulestring, it no longer qualifies.

[confocaloid]

[...] I am simply going off of the rulespace as defined on the respective pages of Snark and Snark64 on the lifewiki (I think there isn't much hope for slight variations that don't affect how snarks work, but I obviously could be wrong). [...]

For the Snark, I think the variant at the top works identically in B3/S23 through B34c8/S234c5e6n8 (2^6 = 64 CA), and each of the other three variants works identically in B3/S23 through B34cq8/S234c5e6n8 (2^7 = 128 CA).

Code: Select all

x = 51, y = 52, rule = B3/S23
20b2o$20bobo$22bo4b2o$18b4ob2o2bo2bo$18bo2bobobobob2o$21bobobobo$22b2o
bobo$26bo2$12b2o$13bo7b2o$13bobo5b2o$14b2o25bo$39b3o$38bo$38b2o3$46b2o
$24b2o21bo$24bo22bob2o$14b3o8b3o11b2o4b3o2bo$4bo11bo10bo11b2o3bo3b2o$
2b5o8bo5b2o21b4o$bo5bo13bo8b2o15bo$bo2b3o12bobo7bobo12b3o$2obo15b2o8bo
13bo$o2b4o21b2o14b5o$b2o3bo3b2o11bo22bo2bo$3b3o4b2o11b3o22b2o$3bo22bo$
2obo21b2o$2ob2o3$11b2o$12bo$9b3o$9bo25b2o$28b2o5bobo$28b2o7bo$37b2o2$
24bo$23bobob2o4b2o$23bobobobo2bo2bo$22b2obobobo3b2o$23bo2b2ob4o$23bo4b
o3bo$24b3obo2bo$26bobobo$29bo!

Ignoring the catalysts, there may be other completions of the base reaction that work in CA not covered by those four Snark variants. See for example a "so near, but yet so far" remark in the forum post viewtopic.php?p=1960#p1960

[...] EDIT: it seems snark works under B4z, but I can't figure out where it differs.

The B4z condition is queried in this generation; the effect is that the catalyst recovers earlier:

Code: Select all

x = 31, y = 25, rule = B34z/S23
3b2o$4bo9bo$2bo11b2o$2b5o6bobo5b2o$7bo13bo$4b3o12bobo$3bo15b2o$3bob2o$
b2o2b2o3b2o$o2b2o6bo$2obo6bo$3bo14b2o$3b2o13bobo$18bo2$11b2o$12bo$9b3o
$9bo4$29b2o$28b2o$30bo!

[...] what constitutes a "Life-like" cellular automota is a very vague term [...]

[...] As soon as you start adding letters in the rulestring, it no longer qualifies.

Life-like cellular automata are a common traditional term for 2-state CA on the square grid, with range-1 Moore neighbourhood, and with rules that depend only on the cell's current state and the number of alive neighbours. That basically means no letters in the rulestring besides 'B' and 'S', no other tessellations, no higher ranges, and just two states (dead / alive).
My suggestion to extend the scope of the thread may or may not be a good idea. With those restrictions, it seems hard to get new omniperiodicity results.

[WhiteHawk]

[...] I am simply going off of the rulespace as defined on the respective pages of Snark and Snark64 on the lifewiki (I think there isn't much hope for slight variations that don't affect how snarks work, but I obviously could be wrong). [...]

For the Snark, I think the variant at the top works identically in B3/S23 through B34c8/S234c5e6n8 (2^6 = 64 CA), and each of the other three variants works identically in B3/S23 through B34cq8/S234c5e6n8 (2^7 = 128 CA).

Code: Select all

x = 51, y = 52, rule = B3/S23
20b2o$20bobo$22bo4b2o$18b4ob2o2bo2bo$18bo2bobobobob2o$21bobobobo$22b2o
bobo$26bo2$12b2o$13bo7b2o$13bobo5b2o$14b2o25bo$39b3o$38bo$38b2o3$46b2o
$24b2o21bo$24bo22bob2o$14b3o8b3o11b2o4b3o2bo$4bo11bo10bo11b2o3bo3b2o$
2b5o8bo5b2o21b4o$bo5bo13bo8b2o15bo$bo2b3o12bobo7bobo12b3o$2obo15b2o8bo
13bo$o2b4o21b2o14b5o$b2o3bo3b2o11bo22bo2bo$3b3o4b2o11b3o22b2o$3bo22bo$
2obo21b2o$2ob2o3$11b2o$12bo$9b3o$9bo25b2o$28b2o5bobo$28b2o7bo$37b2o2$
24bo$23bobob2o4b2o$23bobobobo2bo2bo$22b2obobobo3b2o$23bo2b2ob4o$23bo4b
o3bo$24b3obo2bo$26bobobo$29bo!

Ignoring the catalysts, there may be other completions of the base reaction that work in CA not covered by those four Snark variants. See for example a "so near, but yet so far" remark in the forum post viewtopic.php?p=1960#p1960

[...] EDIT: it seems snark works under B4z, but I can't figure out where it differs.

The B4z condition is queried in this generation; the effect is that the catalyst recovers earlier:

Code: Select all

x = 31, y = 25, rule = B34z/S23
3b2o$4bo9bo$2bo11b2o$2b5o6bobo5b2o$7bo13bo$4b3o12bobo$3bo15b2o$3bob2o$
b2o2b2o3b2o$o2b2o6bo$2obo6bo$3bo14b2o$3b2o13bobo$18bo2$11b2o$12bo$9b3o
$9bo4$29b2o$28b2o$30bo!

[...] what constitutes a "Life-like" cellular automota is a very vague term [...]

[...] As soon as you start adding letters in the rulestring, it no longer qualifies.

Life-like cellular automata are a common traditional term for 2-state CA on the square grid, with range-1 Moore neighbourhood, and with rules that depend only on the cell's current state and the number of alive neighbours. That basically means no letters in the rulestring besides 'B' and 'S', no other tessellations, no higher ranges, and just two states (dead / alive).
My suggestion to extend the scope of the thread may or may not be a good idea. With those restrictions, it seems hard to get new omniperiodicity results.

Ah. Thanks.

With b4z, that make 160+128 = 288 rules which can, in theory, be solved for Omniperiodicity using Snarks/Snark64.

Are there any other transition additions/removals which allow the snark to survive, but change it's evolution?

EDIT: confocaloid, might it still be a good idea to spin off some of this discussion into a new thread where Omniperiodicity in Snark/Snark64's working Rulespace can be more properly explored?

DOUBLE EDIT: Snark64 is also able to survive S6e. I am having trouble getting a proper rulespace, however.

[actinophrys]

Sokwe has helpfully provided a proof that both rule 150 and rule 90 are omniperiodic on segments with always-on or always-off ends. This confirms a lot more life-like cellular automata as omniperiodic. Here are examples I have collected that are not in the list above:

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# Rule 90: B2(78)/S(0)23(4678)
x=16, y=8, rule=B2/S23
bo2bo5bo2bo$3ob8ob2o$3bo8bo$b2o4b2o4bo$2bo4b2o4b2o$3bo8bo$b2ob8ob3o$2bo2bo5bo2bo!

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# Rule 90: B2(78)/S(02)34(678) - closed ends increasing length by 1
x=30, y=8, rule=B2/S34
bo2bo2bo2bo7bo2bo2bo2bo$12o5b12o$bo8bo5b2o10b2o$5b2o10bo4b2o4bo$5b2o10bo4b2o4bo$bo8bo5b2o10b2o$12o5b12o$bo2bo2bo2bo7bo2bo
2bo2bo!

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# Rule 150: B2(78)/S(0)1(24)56(78)
x=16, y=10, rule=B2/S156
2bo10bo$3b10o$ob12obo$b3o8b3o$b2o4b2o4b2o$b2o4b2o4b2o$b3o8b3o$ob12obo$3b10o$2bo10bo!

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# Rule 150: B2(78)/S(0)1(24)5(6)78
x=16, y=12, rule=B2/S1578
4bo6bo$5b6o$2bob8obo$3b10o$ob2o8b2obo$b3o3b2o3b3o$b3o3b2o3b3o$ob2o8b2obo$3b10o$2bob8obo$5b6o$4bo6bo!

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# Rule 90: B4(78)/S(01)246(7)8
x=14, y=8, rule=B4/S2468
2bo8bo$b2o3b2o3b2o$ob10obo$obo3b2o3bobo$b2o8b2o$o12bo$b3o6b3o$4b6o!

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# Rule 150: B4(78)/S(012)3567(8)
x=13, y=13, rule=B4/S3567
2bo$b3o$3o$b3o$2b3o$3b4o$4b4o$5b3o$6b3o$7b3obo$8b5o$9b3o$10bo!

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# Rule 150: B6(8)/S(0)123457(8) - rounded ends increasing length by 1
x=22, y=3, rule=B6/S123457
2ob2ob2o5b2ob2ob2o$o2b2o2bo4bo2bo2bo2bo$b6o6b8o!

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# Rule 90: B(24)7/S(01)23457
x=16, y=7, rule=B247/S23457
bo11bo$2obo8b3o$b2ob8o$2b2o3b2o3b2o$4b8ob2o$b3o8bob2o$2bo11bo!

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# Rule 90: B(25)7/S(01)23457
x=14, y=9, rule=B257/S23457
bo9bo$3o7b3o$bo9b3o$3b7o2bo$3bo2b2o2bo$bo2b7o$3o9bo$b3o7b3o$2bo9bo!

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# Rule 90: B(34)7/S(01)23457 - vertically invert ends to change length by ±1
x=14, y=9, rule = B347/S23457
2bobobobobo$2b9o$3o7b3o$bob7obobo$2obo2b2o2bob2o$obob7obo$b3o7b3o$3b9o$3bobobobobo!

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# Rule 90: B(45)7/S(01)23457
x=8, y=3, rule=B457/S23457
7o$o2b2o2bo$b7o!

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# Rule 150: B(24)7/S(012)34568 - ends with extra polyomino increasing length by 1
x=38, y=13, rule=B247/S34568
bo10bo8bo12bo$3o8b3o6b3o10b3o$bob8obo8bob10obo$3bob2ob2obo11b2ob2ob2o2bo$4bo2bo2b2o13bo2bo2bob3o$2b9o2bo10b9ob3o$b4o2b2o
2b4o7bobo2bo2bo2bobo$2bo2b9o7b3ob9o$4b2o2bo2bo10b3obo2bo2bo$4bob2ob2obo11bo2b2ob2ob2o$3bob8obo8bob10obo$2b3o8b3o6b3o10b3o
$3bo10bo8bo12bo!

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# Rule 150: B(25)7/S(012)34568 - ends with straight projections increasing length by 1
x=38, y=13, rule=B257/S34568
11bo$10b3o$11b2obo$11b5o$4bo2bo2bo3bo10bo2bo2bo$3b8o10bo2b9o3bo$4bo2b2o2bo8b5o2bo2bo2b5o$5b8o8bo3b9o2bo$bo3bo2bo2bo14bo2b
o2bo$5o$bob2o$3b3o$4bo!

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# Rule 150: B(34)7/S(012)34568 - ends with block corners increasing length by 1
x=34, y=11, rule=B347/S34568
b2ob2ob2ob2o8b2ob2ob2obob2o$ob9obo6bob12o$2o2b2ob2o2b2o5bobo2b2ob2obobo$bobo2bo2bobo6b2ob2o2bo2bo2b2o$2ob7ob2o6b2ob9o2bo$
o2bo2b2o2bo2bo4b2obo2bo2bo2bob2o$b2ob7ob2o4bo2b9ob2o$2bobo2bo2bobo6b2o2bo2bo2b2ob2o$b2o2b2ob2o2b2o6bobob2ob2o2bobo$bob9ob
o5b12obo$2b2ob2ob2ob2o6b2obob2ob2ob2o!

[Layz Boi]

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# Rule 150: B2(78)/S(0)1(24)5(6)78
x=16, y=12, rule=B2/S1578
4bo6bo$5b6o$2bob8obo$3b10o$ob2o8b2obo$b3o3b2o3b3o$b3o3b2o3b3o$ob2o8b2obo$3b10o$2bob8obo$5b6o$4bo6bo!

Here's a Rule 150 emulator for B2(78)/S(0)1(24)5(6)7(8) which subsumes the rules that one works in.

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x = 42, y = 16, rule = B2/S157
6bo3bo20bo3bo$3bo3b3o3bo14bo3b3o3bo$4b9o3bo8bo3b9o$bob4o3b6o10b6o3b4ob
o$2b4o3bo3b16o3bo3b4o$2b3o7bo3b10o3bo7b3o$ob2o11bo10bo11b2obo$b2o15bob
2obo15b2o$b2o15bob2obo15b2o$ob2o11bo10bo11b2obo$2b3o7bo3b10o3bo7b3o$2b
4o3bo3b16o3bo3b4o$bob4o3b6o10b6o3b4obo$4b9o3bo8bo3b9o$3bo3b3o3bo14bo3b
3o3bo$6bo3bo20bo3bo!

[actinophrys]

Good catch that S8 isn't needed there. I'd also missed this very simple case:

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# Rule 150: B2(78)/S(0)1(2)45(678)
x=10, y=10, rule=B2/S145
o8bo$b8o$b8o$o8bo$4b2o$4b2o$o8bo$b8o$b8o$o8bo!

Edit Feb 26: On the discord we also found this emulator with ends by sylvani and Layz Boi. That should finish all the cases where this kind of domino track might work and there are p1s including components with more than two cells to make the stator.

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# Rule 90: B2(78)/S0(2)3(4678) - symmetrical ends increasing length by 1
x=32, y=20, rule = B2/S03
8bo17bo$2bo$6bob2o15b2obo$ob2o4b2obo11bob2o$2b2ob2o21b2o$5b2o2bo15bo2b2obo$ob2o4b3o2bo2bo2bo4b3o$2b2obobo3b13o3bobo$7b2ob
o11bo3b2o$3bo3bo2b2o4b2o9bo$6bo3bo5b2o9bo$2b2ob2o3bo11bo3b2o$ob2o2bo3b14o3bobo$7b3o3bo2bo2bo4b3o$2bob2o2bo16bo2b2obo$4b2o
22b2o$7b2obo12bob2o$5bob2o16b2obo2$7bo18bo!

There are also these B2 emulators where the stator is a row of spaced cells inside a container:

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# Rule 90: B2(8)/S024(5)6(78) - ends with hexominoes increasing length by 1
x=20, y=13, rule = B2/S0246
16bo$4bo9bob2o$16b2obo$2bob11o2bo$ob2o11bo$4bo4b2o4bo$2bobobo2bo2bo2bo$4bo4b2o4bo$ob2o11bo$2bob11o2bo$16b2obo$4bo9bob2o$
16bo!

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# Rule 150: B2(8)/S(0)13456(78)
x= 14, y = 7, rule = B2/S13456
o2bo2bo2bo3bo$b12o$bo10bo$2o3bo2bo3b2o$bobo2bo2bo2bo$b12o$o12bo!

For other types, qqd found a neat rule 150 emulator in B4(678)/S(01)235(678). Then Sokwe had mentioned their proof should also work for rule 105 and 165. Emulators for those seem to be less common, but here are some that I found:

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# Rule 165: B3(6)7(8)/S(01)2345(7)8
x=22, y=10, rule=B3678/S23458
3b2o10b2obo$3b2o10bob2o$o6bo4bo8bo$8o4b10o3$8o4b10o$o6bo4bo8bo$3b2o10b2obo$3b2o10bob2o!

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# Rule 105: B(2)68/S(01)23457
x=14, y=7, rule=B268/S23457
bo10bo$3o8b3o$bob8obo$bo10bo$bob8obo$3o8b3o$bo10bo!

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# Rule 165: B(245)68/S(012)34568 - more indented ends increasing length by 1
x=40, y=15, rule = B24568/S34568
4bo7bo11bo10bo$3b3o5b3o9b3o8b3o$2b2ob2o3b2ob2o7b2ob2o6b2ob2o$b2ob2ob3ob2ob2o5b2ob2ob6ob2ob2o$15o5b20o$bobobo2bo2bobo7bobo
bo2bo2bo2bobobo$2b2ob8o9b2ob10ob2o$2b4o6bo9b4o8b4o$2b2ob8o9b2ob10ob2o$bobobo2bo2bobo7bobobo2bo2bo2bobobo$15o5b20o$b2ob2ob
3ob2ob2o5b2ob2ob6ob2ob2o$2b2ob2o3b2ob2o7b2ob2o6b2ob2o$3b3o5b3o9b3o8b3o$4bo7bo11bo10bo!

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# Rule 165: B(34)68/S(012)34568 - rounded ends increasing length by 1
x=14, y=9, rule = B3468/S34568
2ob2obobob2o$12o$bobo2bo2bob3o$2ob8ob2o$b3o6b3o$2ob8ob2o$bobo2bo2bob3o$12o$2ob2obobob2o!

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# Rule 165: B(35)68/S(012)34568 - smaller ends increasing length by 1
x=18, y=6, rule=B3568/S34568
2b2obo2bo2bo$3ob10o$5o8b3o$2bob10ob3o$4obo2bo2bo2b4o$2o12b2o!