One-cell growth per generation

[lukebradford]

I was intrigued by this pattern by Nicolay Beluchenko, a tubstretcher that grows by exactly one cell per generation:

Code: Select all

x = 17, y = 15, rule = B3/S23
8b2o7b$7b2o8b$9bo7b$11b2o4b$10bo6b2$9bo2b2o3b$b2o5b2o4bo2b$2o5bo5bo3b$
2bo4bobo3b2o2b$4bo2bo4b2obob$4b2o7b2o2b$8bo4bob2o$7bobo2bob2ob$8bo!

I thought this might be an interesting puzzle to work on in other CAs. In Live Free or Die (b2/s0), it's easy, because there's a small, common p4 puffer that produces one cell per period. With four of these, one in each phase, you're done:

Code: Select all

x = 10, y = 30, rule = B2/S0
8bo$4bo4bo$5bo3bo$7bo$6bo4$8bo$4bo4bo$3bobo3bo$4bo2bo$6bo4$8bo$2bobo4b
o$5bo3bo$2bo4bo$4bobo4$2bo5bo$b2obo4bo$2obobo3bo$b3o3bo$2b3obo$4bo!

(Of course there may be smaller ways to do this.)

Please post any one-cell-per-generation patterns you have for other rules, or improvements to these! Are there any Life-like CAs that seem particularly suited to this task?

[Alexey_Nigin]

In Live Free or Die (b2/s0), it's easy, because there's a small, common p4 puffer that produces one cell per period.

In 2x2, it is also very easy, as there is a puffer with the same property:

Code: Select all

x = 6, y = 78, rule = B36/S125
5bo$4bo2$2b2o$2b2o$2bo$bob2o$2b3o3$5bo$4bo2$3bo$bobo2$bo2bo$2bobo3$5bo
$4bo2$3bo$2bo$b2o$2b3o$3bo3$5bo$4bo2$3bo$obo2$2b3o4$5bo$4bo2$b2o$b2ob
2o$bob3o$2bo4$5bo$4bo2$b2o$4b2o$bo$2bobo4$5bo$4bo2$2bo$b2o2bo$2b3o$3bo
4$5bo$4bo2$2bo$2bo2bo$2bobo2$4bo!

[Alexey_Nigin]

Here is a small example in B2/S234:

Code: Select all

x = 9, y = 10, rule = B2/S234
2bo$o$o$2bo4bo$3b6o$7bo$4bo$2bo$2bo$4bo!

[thunk]

In 2x2, it is also very easy, as there is a puffer with the same property:

Code: Select all

x = 6, y = 78, rule = B36/S125
5bo$4bo2$2b2o$2b2o$2bo$bob2o$2b3o3$5bo$4bo2$3bo$bobo2$bo2bo$2bobo3$5bo
$4bo2$3bo$2bo$b2o$2b3o$3bo3$5bo$4bo2$3bo$obo2$2b3o4$5bo$4bo2$b2o$b2ob
2o$bob3o$2bo4$5bo$4bo2$b2o$4b2o$bo$2bobo4$5bo$4bo2$2bo$b2o2bo$2b3o$3bo
4$5bo$4bo2$2bo$2bo2bo$2bobo2$4bo!

Use the double wickstretcher--half the puffers required:

Code: Select all

x = 8, y = 37, rule = B36/S125
5bo$4bo2$2b2o3bo$2b2o2bo$2bo$bob2o$2b3o3$5bo$4bo2$3bo3bo$bobo2bo2$bo2b
o$2bobo3$5bo$4bo2$3bo3bo$2bo3bo$b2o$2b3o$3bo3$5bo$4bo2$3bo3bo$obo3bo2$
2b3o!

[Citation needed]

I've been looking through some old and short articles on LifeWiki, and the "One per generation" caught my attention. I decided that this topic is a perfect one to test the Life enthusiasts' creativity.

Why choose 1 Cell Per Generation? Some topics such as fuses and pure glider generators are not notable enough, while others such as conduits, syntheses, crawlers and unit cells focus too much on macroscopic properties or computational uses. Quadratic growth patterns are rather difficult to construct: It took ColorfulGalaxy four days to complete the switch engine ping pong synthesis. Linear growth patterns are much easier to construct, and can usually be found using apgsearch or other search programs. Some of the linear growth patterns grow too slowly, so you may have to use multiple copies of them along with some oscillators. Others grow too fast, and may have to be monomerized.

Prove empirically. We are not just trying to prove that such patterns exist. We are explicitly constructing them. In other words, we are looking for a constructive proof that such patterns exist in given rules.

Be creative. Different types of linear growth can be used, such as guns, puffers, rakes, wickstretchers, wavestretchers and so on.

Show the process. You can tell us how you gradually completed the "One per generation" patterns, how you found your objects needed, how you were inspired, why you think your patterns are interesting, which program or script you used, and so on.

Today is the holiday for programmers. To celebrate the holiday, ColorfulGalaxy and his friends assembled these patterns that grow by one cell per generation. Please read the article on One per generation.

This pattern works in Conway's Game of Life, while the pattern on LifeWiki fails in the rule below.

Code: Select all

#C One Cell Per Generation
x = 306, y = 55, rule = B37c8/S238
90bo$90b3o144bo$82bo10bo143b3o$82b3o3b3o2bo2b2o55bo75bo10bo$85bobobob
obobobo55b3o73b3o3b3o2bo2b2o$80b3o2bobo3bobobo60bo75bobobobobobobo$79b
obobobobobobobob2o54b3o2bo2b2o66b3o2bobo3bobobo$75b2o2bo3bobobobo3bo56b
obobobobobo65bobobobobobobobob2o$75bobobobobobobobobobo52b2o2bo3bobob
o63b2o2bo3bobobobo3bo$77bobobo3bobo2b3o53bobobobobobob2o62bobobobobob
obobobobo$76b2obobobobobo60bobobo3bo67bobobo3bobo2b3o$79bo2b3o3b3o56b
2obobobobo66b2obobobobobo$79bo10bo59bo2b3o70bo2b3o3b3o$80b3o67bo75bo10b
o$82bo68b3o73b3o$153bo75bo4$173b2o3b2o$106bo20b2o3b2o39bobobobo$65bo40b
3o18bobobobo40bo3bo$65b3o30bo10bo18bo3bo$57bo10bo29b3o3b3o2bo2b2o59b2o
3b2o118bo$34bo22b3o3b3o2bo2b2o28bobobobobobobo13b2o3b2o34b2o2bobo3bob
o2b2o28bo84b3o$34b3o23bobobobobobobo23b3o2bobo3bobobo10b2o2bobo3bobo2b
2o11b2o3b2o11bobob2o5b2obobo28b3o35bo38bo10bo$26bo10bo17b3o2bobo3bobo
bo24bobobobobobobobob2o9bobob2o5b2obobo13bobo14bo13bo21bo10bo34b3o36b
3o3b3o2bo2b2o$26b3o3b3o2bo2b2o12bobobobobobobobob2o19b2o2bo3bobobobo3b
o13bo13bo11bo2bobo2bo47b3o3b3o2bo2b2o22bo10bo38bobobobobobobo$29bobob
obobobobo8b2o2bo3bobobobo3bo22bobobobobobobobobobo38bo3bobo3bo10bo13b
o24bobobobobobobo22b3o3b3o2bo2b2o29b3o2bobo3bobobo$24b3o2bobo3bobobo10b
obobobobobobobobobo24bobobo3bobo2b3o14bo13bo10bo3bobo3bo9bobob2o5b2ob
obo18b3o2bobo3bobobo27bobobobobobobo28bobobobobobobobob2o$23bobobobob
obobobob2o11bobobo3bobo2b3o24b2obobobobobo18bobob2o5b2obobo13bobo13b2o
2bobo3bobo2b2o17bobobobobobobobob2o21b3o2bobo3bobobo26b2o2bo3bobobobo
3bo$19b2o2bo3bobobobo3bo13b2obobobobobo32bo2b3o3b3o15b2o2bobo3bobo2b2o
8bobob2ob2obobo13b2o3b2o18b2o2bo3bobobobo3bo23bobobobobobobobob2o25bo
bobobobobobobobobo$19bobobobobobobobobobo16bo2b3o3b3o29bo10bo20b2o3b2o
15bob2ob2obo40bobobobobobobobobobo19b2o2bo3bobobobo3bo30bobobo3bobo2b
3o$21bobobo3bobo2b3o17bo10bo30b3o75bo3bo21bobobo3bobo2b3o20bobobobobo
bobobobobo29b2obobobobobo$20b2obobobobobo23b3o40bo29bo3bo40bobobobo19b
2obobobobobo27bobobo3bobo2b3o33bo2b3o3b3o$23bo2b3o3b3o22bo69bobobobo39b
2o3b2o22bo2b3o3b3o23b2obobobobobo38bo10bo$23bo10bo92b2o3b2o68bo10bo26b
o2b3o3b3o36b3o$24b3o176b3o34bo10bo38bo$26bo178bo35b3o$243bo$130b2o$129b
3o158b2o$117bo8bob2o9b2o149b3o$80bo36bobo6bo2bo9b2o135bo15b2obo5b2o$79b
2o37bobo5bob2o109bo34bobo15bo2bo5b2o$26bo36b2o13b2o8b2o15b2o11bo2bo7b
3o107b2o32bobo16b2obo$26bobo34b3o11b3o8b2o15b2o11bobo9b2o53bo41b2o11b
2o8b2o15b2o3bo2bo14b3o$9bo17bobo4b2o29b2obo9b2o37bobo63bobo40b3o11b3o
7b2o15b2o4bobo14b2o$9b2o16bo2bo3b2o18b2o9bo2bo10b2o36bo56bo7bobo11b2o
25bob2o13b2o32bobo$10b2o15bobo24b2o9b2obo11bo92b2o6bo2bo11b2o18b2o5bo
2bo12b2o35bo$2o8b3o13bobo34b3o106b2o4b2o2bobo31b2o5bob2o12bo$2o8b2o14b
o36b2o97b2o7b3o4b2o3bobo40b3o$9b2o151b2o8b2o4b2o5bo41b2o$9bo163b2o$174b
o!

This one works in "tlife" and is based on guns.

Code: Select all

x = 1564, y = 175, rule = B3/S2-i34q
715bo7bo85bo7bo26bo7bo$715bo7bo18bo7bo24bo7bo25bo7bo26bo7bo$714bobo5b
obo17bo7bo24bo7bo24bobo5bobo24bobo5bobo$679bo7bo53bobo5bobo22bobo5bob
o$679bo7bo27bo3bo3bo85bo3bo3bo26bo3bo3bo$678bobo5bobo28bo3bo20bo3bo3b
o24bo3bo3bo27bo3bo30bo3bo$717bo3bo22bo3bo28bo3bo29bo3bo30bo3bo26bo7bo
$679bo3bo3bo29bobobo22bo3bo28bo3bo29bobobo30bobobo26bo7bo$681bo3bo32b
obo23bobobo28bobobo30bobo32bobo26bobo5bobo$681bo3bo59bobo30bobo$681bo
bobo191bo3bo3bo$682bobo194bo3bo$879bo3bo$879bobobo$880bobo5$720b2o92b
2o33b2o$719bo2bo24b2o31b2o31bo2bo31bo2bo$718bob2o24bo2bo29bo2bo29bob2o
31bob2o$684b2o31bo2b2o23bob2o29bob2o29bo2b2o30bo2b2o$683bo2bo29bob2o24b
o2b2o28bo2b2o28bob2o31bob2o$682bob2o30bob2o23bob2o29bob2o30bob2o31bob
2o$681bo2b2o31bo25bob2o29bob2o31bo34bo35b2o$680bob2o60bo32bo103bo2bo$
680bob2o196bob2o$681bo197bo2b2o$878bob2o$878bob2o$879bo3$719bo3b2o88b
o3b2o29bo3b2o$718bobo2b2o21bo3b2o27bo3b2o27bobo2b2o28bobo2b2o$718bob3o
22bobo2b2o26bobo2b2o27bob3o30bob3o$683bo3b2o28b2o4b2o20bob3o28bob3o28b
2o4b2o27b2o4b2o$682bobo2b2o27bo3b2o3bo18b2o4b2o25b2o4b2o25bo3b2o3bo25b
o3b2o3bo$682bob3o30b2obob2o2bo16bo3b2o3bo23bo3b2o3bo25b2obob2o2bo25b2o
bob2o2bo$681b2o4b2o29bo3bob2o18b2obob2o2bo23b2obob2o2bo25bo3bob2o27bo
3bob2o26bo3b2o$680bo3b2o3bo27bo2b2o2bo20bo3bob2o25bo3bob2o25bo2b2o2bo
27bo2b2o2bo26bobo2b2o$681b2obob2o2bo26b4o3bo19bo2b2o2bo25bo2b2o2bo26b
4o3bo27b4o3bo26bob3o$682bo3bob2o25b2o4b2ob2o18b4o3bo25b4o3bo24b2o4b2o
b2o24b2o4b2ob2o24b2o4b2o$681bo2b2o2bo27bob2obobo18b2o4b2ob2o22b2o4b2o
b2o24bob2obobo27bob2obobo25bo3b2o3bo$681b4o3bo26bo2b2obobo19bob2obobo
25bob2obobo25bo2b2obobo26bo2b2obobo26b2obob2o2bo$679b2o4b2ob2o25b2o3b
obo19bo2b2obobo24bo2b2obobo25b2o3bobo27b2o3bobo28bo3bob2o$680bob2obob
o32bo21b2o3bobo25b2o3bobo31bo34bo29bo2b2o2bo$679bo2b2obobo31b2o26bo32b
o32b2o33b2o29b4o3bo$679b2o3bobo59b2o31b2o96b2o4b2ob2o$684bo193bob2obo
bo$683b2o192bo2b2obobo$877b2o3bobo$882bo$881b2o$724b2o92b2o33b2o$724b
o26b2o31b2o32bo34bo$718b2ob2obo26bo32bo27b2ob2obo28b2ob2obo$688b2o24b
o2bobobobob2o18b2ob2obo26b2ob2obo23bo2bobobobob2o22bo2bobobobob2o$688b
o25b4obobobobo15bo2bobobobob2o20bo2bobobobob2o21b4obobobobo23b4obobob
obo$682b2ob2obo30bo2b2obo15b4obobobobo21b4obobobobo27bo2b2obo28bo2b2o
bo$678bo2bobobobob2o25b3o5bo21bo2b2obo26bo2b2obo24b3o5bo26b3o5bo32b2o
$678b4obobobobo26bo2b4o20b3o5bo24b3o5bo25bo2b4o28bo2b4o34bo$683bo2b2o
bo27bo2bo2b4o16bo2b4o26bo2b4o28bo2bo2b4o25bo2bo2b4o24b2ob2obo$680b3o5b
o26bob2o4bo2bo17bo2bo2b4o23bo2bo2b4o22bob2o4bo2bo23bob2o4bo2bo20bo2bo
bobobob2o$680bo2b4o27bobo25bob2o4bo2bo21bob2o4bo2bo21bobo32bobo30b4ob
obobobo$681bo2bo2b4o23bobo24bobo30bobo31bobo32bobo35bo2b2obo$679bob2o
4bo2bo24bo25bobo30bobo32bo34bo33b3o5bo$678bobo61bo32bo102bo2b4o$678bo
bo198bo2bo2b4o$679bo197bob2o4bo2bo$876bobo$876bobo$877bo5$725b2o92b2o
33b2o$725bo26b2o31b2o32bo34bo$719b2ob2obo26bo32bo27b2ob2obo28b2ob2obo
$689b2o24b2obobobobob2o18b2ob2obo26b2ob2obo23b2obobobobob2o22b2obobob
obob2o$689bo25bob2obo3bobo15b2obobobobob2o20b2obobobobob2o21bob2obo3b
obo23bob2obo3bobo$683b2ob2obo30bobobo2bo14bob2obo3bobo21bob2obo3bobo27b
obobo2bo27bobobo2bo$679b2obobobobob2o24b2obo5b2o20bobobo2bo25bobobo2b
o22b2obo5b2o24b2obo5b2o31b2o$679bob2obo3bobo25bo3b4o19b2obo5b2o22b2ob
o5b2o23bo3b4o27bo3b4o34bo$684bobobo2bo25b2obo2bob2o16bo3b4o25bo3b4o27b
2obo2bob2o25b2obo2bob2o25b2ob2obo$680b2obo5b2o28bo4bobo17b2obo2bob2o23b
2obo2bob2o26bo4bobo27bo4bobo21b2obobobobob2o$680bo3b4o28bobo27bo4bobo
25bo4bobo23bobo32bobo29bob2obo3bobo$681b2obo2bob2o25b2o25bobo30bobo31b
2o33b2o35bobobo2bo$683bo4bobo52b2o31b2o100b2obo5b2o$680bobo195bo3b4o$
680b2o197b2obo2bob2o$881bo4bobo$878bobo$878b2o11$779b4o$778bo4bo$778b
2o2b2o$778bo4bo$778bob2obo$777bo6bo$777bo6bo$778bob2obo$778bo4bo$778b
2o2b2o$778bo4bo$779b4o13$115bo$116bo$115bobo$110b3o4b2o$9bo100b3o3bob
o$8bobo99b2o3b4o382b3o418b3o$8bob2o99bob2ob2o188bo193b2ob2o416b2ob2o89b
o$9b2o99b3ob2o188bo2b4o188bo2b3o416b2ob3o86bo2bo$59bo51b2ob2o188bo193b
obo2bob2o412b2ob2obo46bo39bo3b2o$58bob2o50b3o48b2o139bo199b4o410b4o3b
2o44b4o37bo3bo$57b5o102bo39b2obo97b2o197bo2bo410bobo3b3o44bo3bo35bo2b
o2bo98bo3b2o$56b2obo2bo99b3o38bo2bo148b2o42bo101b2o2b3o91bo97bo2bo119b
2o96b2o4b3o44b5o36b2o2b3o103bo$57b2ob2o101bo41bo2bo146bo42b3o100b6o90b
2ob2o95bo121b2obo96bobo49bo40bo2bo46b2obo56bo$57b2ob2o97b5o38b3o2b2o45b
o142bo2bo100b3obo45b3o41bo2bo3bo94bo122bobo97bo49b3o41bo2bo45b2o51b4o
2bo192bob3o$59bo99bo3bo39bo2bo2bo43b2o99bobo41b2o48bo102bob5o38bo2bo48b
o48bo123bo47bo51bo48bo42bob2o103bo193b6o$160b4o40bo3bo44bobo98bo91b2o
b2o102bobob2o36bobo5bo44bo47bo3bo166b2ob2o98b2o86b2o110bo142b3o2b2o191b
3o$162bo40b2o3bo42bo2bo191b2ob2o104bobo37bo2bob3o44bo52bo166b2ob2o102b
o85bo4b2ob2o98bobo44b2o96bo2bo193bo$204bo2bo42bob2o191bob2o2bo97b2o4b
obo38bob2obo95b3o166bo2bob2o101bo90b2obo145bo2bo95b4o193bo3bo$205bo43b
2ob2o47bo144b2ob2o104b2o41bo52bo215b5o192bo2bo101bo44b3o97b2obo2bobo39b
2o42bo106bo$300bo2bo143b4o193bo5b2o93bo2bo118b2obo192bobo102b2o45bo100b
3o2bo39bobo4b2o33bob2obo104bo$299b2o3bo143bo193b2obo4bo94b2o122bo194b
2o198bo51b2ob2o40bobo38b3obo2bo103bo$3bo250b2o44bo3bo338b2ob2o3bo96b2o
314bo197b4o51b3o40b2obobo35bo5bobo45bo54bo2bo45b2o$3bo249bob2o42bo2bo
2bo340b2o269bo344b2ob2o94b5obo37bo2bo46b2o105b2o$3bo101b2o191b3o2b2o340b
o270bobo342bo2b2obo97b3o34bo3bo2bo47bo102bo2bo$3bo100bob2o89b3o101bo2b
o44bob2o293b2o267b2obo343b2ob2o136b2ob2o47b2o$o3bo99bobo89b2ob2o98bo2b
o47b2o295bo165b3o100b2o88bo255b2ob2o138bo45bo3b2ob2o48bo$o104bo49bo39b
3ob2o99b2obo512bo148bo39bo258bo187bo4bob2o48bo$b3o149b2ob2o38bob2ob2o
44bo564bo3bo146b2obo37bobo444b2o5bo48b3o$153b2ob2o37b2o3b4o41b4o96b2o
b2o144b2o317bo149b5o35b2o4b3o97bob2o338bo$50bo2bo98b2obo2bo36b3o3bobo
40bo3bo97bob2o42b2o3bo95bo2bo316bo148bo2bob2o34bobo3b3o98bo2bo197bo189b
o$52b2o99b5o37b3o4b2o40b5o98bo2bo40bo102b3o197bo118bo149b2ob2o35b4o3b
2o96bo2bo198b3o89b3o50bo43b2ob2o$49b2o103bob2o42bobo45bo100bobo39bo103b
o196bob2obo115bo2bo47b2o97b2ob2o36b2ob2obo97b2o2b3o45bo46bo102bo2bo87b
2ob2o47bo44bo3bo2bo$155bo45bo45b3o99b2o40bo2b4o146bo41b3obo100bo2bob3o
162b2o102bo40b2ob3o47bo47bo2bo2bo46b2o41b4o2bo101b2o47bo40b3o2bo46b2o
46bo2bo45bo6bobo$200bo48bo100bo42bo149b4o39b6o99bobo5bo43bo117bo2bo141b
2ob2o48bo48bo3bo46bobo47bo148b2ob2o36b2obo2bobo44b2o43bo5bobo44bo7b2o
$248b2o191bo100b2ob2o39b2o2b3o99bo2bo46b2o263b3o45b2o51bo3b2o46bo2bo45b
o148b2ob2o35b4o96b3obo2bo45bo6bo$441bobo97bob2o2bo41bo2bo98bo2bo3bo42b
o313bo53bo2bo48b2obo38bo3b2o148bo2b2obo34bo2bo97bob2obo$542b2ob2o42b4o
100b2ob2o44b2o311b3o53bo44bo4b2ob2o89b2o101b2ob2o35b3o2b2o97bo44bo$442b
o99b2ob2o36bobo2bob2o48b2o53bo43b2ob2o3bo308bo97b2o100bo101b4o37b6o142b
2o5bo$442b2o100bo39bo2b3o44b2o4bobo95b2obo4bo309b5o299bo39bob3o45b3o95b
o4bob2o$585b2ob2o50bobo97bo5b2o308bo3bo98b2o94bobo188b5obo94bo3b2ob2o
$586b3o49bobob2o103bo308b4o98b2obo95bo187b2obobo100b2o$636bob5o414bo388b
obo104bo$636b3o104bo702bobo4b2o96b2o$744bo702b2o102bo$744bo!

Besides the engineered ones above, there are also elementary ones.

Code: Select all

x = 3, y = 3, rule = B1e/S012345678
2o$3o$2o!

Such patterns are also present in multistate rules.

Code: Select all

#C One cell per generation
x = 15, y = 4, rule = Wilfred
13.2C$C.C4.CAC2.C$BC6.CB$CAC.C2.C.C.2C.C!

Code: Select all

#C This pattern cannot be rotated.
x = 3, y = 17, rule = JvN29
LO$IpAI2$PK$IpAI2$LK$MpAI2$LK$IpBI2$LK$IpCI2$LO$IpAM!

[Citation needed]

ColorfulGalaxy sent a signal just now, saying that a new one was found.

Code: Select all

#C VesselLife
x = 6, y = 17, rule = B2n3-q5y6c/S23-k
4bo$4b2o$3bobo$bo$obo$bo6$4b2o$3bobo$5bo$b2o$obo$bo!

[Citation needed]

This rule has two simple duplicators. ColorfulGalaxy needed a p5 oscillator, but there is only one such oscillator known — and he still managed to complete the pattern.

Code: Select all

# Glitter^2 1 cell per generation
# 8 barrels + 5 identical oscillators
x = 11, y = 102, rule = B3acijn4cjyz5cy6an8/S2-n3-ckqy4aw5y6c7c8
4b2o$4b2o2$6bo$5bo$5b3o4$4b2o$4b2o4$4bobo$4b2o$5bo2$4b2o$4b2o6$2b2o$b
2o$4b2o$4b2o4$2bo$bobo$o2bo$b2o$4b2o$4b2o15$3b2ob2o$2o2bobo2b2o$o3bob
o3bo$2b2o3b2o$b3o3b3o7$3b2ob2o$2o2bobo2b2o$o3bobo3bo$2bobobobo$bobo3b
obo$2bo5bo7$3b2ob2o$2o2bobo2b2o$o3bobo3bo$2bobobobo$bobo3bobo$2bo5bo6$
3b2ob2o$2o2bobo2b2o$o3bobo3bo$2b3ob3o$bo7bo$2bo5bo6$3b2ob2o$2o2bobo2b
2o$o3bobo3bo$2b3ob3o$b2o5b2o!

[H. H. P. M. P. Cole]

It is trivial to make one-per-generation patterns (and by extension, n-per-generation patterns where n is some natural number) using guns or wickstretchers. Like you showed, many such patterns are clunky almost to the point of being contrived.

A better use of this thread, arguably, would be to compile small one-per-generation patterns which use at most one infinite growth mechanism (most likely a wickstretcher engine).

Edit: One-per-generation (population is 34+n in generation n):

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x = 28, y = 6, rule = R3,C2,S2,B3,N+
22bo2b3o$14bo3bob2o$bobobobobo2bobo2bo$obobo2bobobo3b2o$o14bobobobo$b
o20bo2b3o!

[confocaloid]

[...] small one-per-generation patterns which use at most one infinite growth mechanism (most likely a wickstretcher engine). [...]

By "at most one infinite growth mechanism", do you really mean at most one unique growth mechanism (although as many instances of that mechanism as desired), or at most one engine?
As far as I can tell, "at most one" can be simplified to "one".

One question is: what is the lowest starting term X of a population sequence (X, X+1, X+2, ...)
In other words, what is the lowest X such that there exists a growing pattern with population(T) = X + T, for T = 0, 1, 2, ...?

In a two-dimensional two-state isotropic cellular automaton, X must be at least 2.
(X = 0 is impossible, because a single alive cell cannot spontaneously form in empty space.
X = 1 is impossible, because there is no way for a single alive cell to evolve into a two-bit pattern one tick later.
For example, assuming the square tiling and the Moore neighbourhood, the only possibilities for population 1 tick later are 0, 1, 4, 5, 8, 9.
Assuming the hexagonal tiling and the honeycomb neighbourhood, the only possibilities for population 1 tick later are 0, 1, 6, 7.)

In a 2D two-state isotropic CA on the triangular tiling, there might be some way to have X = 2, because there exist asymmetric two-bit patterns on the triangular tiling:

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x = 12, y = 6, rule = //3L
2.B7.B$.A9.B$6.A3$5.B.B!
#C [[ GRID VIEWONLY ]]

In a 2D two-state isotropic CA on the square tiling, or on the hexagonal tiling, X = 2 is impossible, because every two-bit pattern remains unchanged when rotated 180 degrees, and therefore any growing pattern of this kind will necessarily grow in at least two directions at once, contradicting "one bit per tick".
Is X = 3 possible?
Is X = 3 possible with range-1 neighbourhoods (the Moore neighbourhood on the square tiling; the honeycomb neighbourhood on the hexagonal tiling)?
Is X = 2 indeed possible on the triangular tiling?

[Citation needed]

Are polyglot "1 cell per generation" (such patterns should work in as many rules as possible) patterns notable?
ColorfulGalaxy discovered one that is "B7c agnostic" (the first pattern I've posted) (see Saka's dictionary), while the pattern on LifeWiki isn't.
Also, this task would be especially hard if there are few or no known oscillators to make up for the bends in the population graph.

[confocaloid]

If by "notable" you mean LW:NB, then no. Most alien reactions/patterns/objects/cellular automata aren't notable.
Polyglots in particular aren't notable just because it's a polyglot. Lots of reactions proceed identically in two or more different CA, and expanding investigation to allow other types of cellular automata can even transform non-polyglots into polyglots.

If by "notable" you mean "worthy of discussion" that depends. If there are really many "optional" rules in the ruletable and/or the reaction is otherwise interesting in some way (i.e. there is something of interest to others to discuss in the first place).

Being or not being an one-bit-per-tick growth doesn't really affect the above.

[H. H. P. M. P. Cole]

[...] small one-per-generation patterns which use at most one infinite growth mechanism (most likely a wickstretcher engine). [...]

By "at most one infinite growth mechanism", do you really mean at most one unique growth mechanism (although as many instances of that mechanism as desired), or at most one engine?
As far as I can tell, "at most one" can be simplified to "one".
[...]

I mean at most one engine.

[confocaloid]

This thread is a duplicate of an earlier forum thread:
viewtopic.php?f=11&t=1410 "One-cell growth per generation"
The threads should be merged.

[Citation needed]

This thread is a duplicate of an earlier forum thread:
viewtopic.php?f=11&t=1410 "One-cell growth per generation"
The threads should be merged.

It's strange that the thread that you mentioned did not appear in Search Results. This may be meant to prevent necroposting.

ColorfulGalaxy discovered these today.

Here is a pattern in FWKnightShip's rule.

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x = 8, y = 5, rule = SoManyShips2
.A3.BAB$ABA2.ABA$6.B$.B4.B$.B4.B!
SMS2 1CPG

"One cell per generation" patterns are fairly simple in Generations rules with B2 but not B1, B0, S2, S1 or S0 (such rules include B2/Sg3 and Star Wars) due to a "wavestretcher".

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x = 3, y = 3, rule = B2/Sg3
.2A$A2B$AB!
This pattern works in many rules.

The previous 7-cell pattern can be reduced to 6 cells.

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#C The pattern can still work even if the cell in the middle is removed.
x = 3, y = 3, rule = B1e/S012345678
2o$obo$2o!

EDIT: 5 cells are enough in this rule.

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x rule = B1e/S012345678
2o$3o!

[FWKnightship]

3-cell pattern that grow by 1 cell per generation:

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x = 3, y = 2, rule = B2a3n/S01c2c
2bo$obo!

[Citation needed]

Here is one in BokaBB's rule.

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#C Traffic Flow 1CPG
x = 94, y = 11, rule = B2e3aciy4jnw68/S23678
bo5bo12b2o7b2o2b2o12b2o2b2o12b2o2b2o12b2o2b2o$bobobobo6b2o4b2o7b2o2b2o
12b2o2b2o12b2o2b2o12b2o2b2o$b3ob3o6bo3bo5bo2bo8bo2bo2bo2bo8bo2bo2bo2b
o8bo2bo2bo2bo8bo2bo$18b2o4b3ob2ob2ob2ob3o2b6ob2ob6o2b5o2b2o2b5o2b5o2b
2o2b5o$b2o3b2o4b2o2b2o11bob2obo14b2o13bo2b2o2bo10bo2b2o2bo$b2o3b2o4b2o
bobo7bob3o4b3obo4bob3o4b3obo4bob2o6b2obo4bob2o6b2obo$16bo3bo4b2o10b2o
4b2o10b2o4b2o10b2o4b2o10b2o$19b2o7b2o4b2o10b2o4b2o10b2o4b2o10b2o4b2o$
2o5b2o7b2o10b2o4b2o10b2o4b2o10b2o4b2o10b2o4b2o$4ob4o7b2o$b3ob3o!

This rule has a very small number of known "p5" oscillators. ColorfulGalaxy tried to use the wickstretcher, but unfortunately it grew too fast, and he didn't manage to monomerize it. He chose the "p10" gun instead.

[pifricted]

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x = 5, y = 4, rule = B2aei3e/S
bobo$o3bo$4bo$3bo!

[Citation needed]

The tubstretcher from Conway's Game of Life actually works in "tlife", but the LWSS doesn't. The "7P4H1V1" spaceship can solve the problem.

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#C tlife 1CPG based on Crab-based tubstretcher
x = 15, y = 38, rule = B3/S2-i34q
8bo$7b2o$7bobo2$10b2o$10b2o2$bo6b2o3bo$2o5bo2bobobo$obo3b2obo3bo$4bo$
3bo$3bo3bo$6bobo$7bo4$b3o$b3o$o$bo5$b3o$b3o$o$bo5$b3o$b3o$o$bo!

Patterns that grow by 1 cell per generation also exist in non-isotropic MAP rules.

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#C Simple MAP 1CPG
x = 1, y = 1, rule = MAPTw8PDw
o!

ColorfulGalaxy has also created a custom rule table that features a "1CPG" pattern growing in a spiral manner.

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#C Custom Rule Table 1CPG
x rule = 1cpg
oB!
@RULE 1cpg

@TABLE
n_states:3
neighborhood:moore
symmetries:rotate4

var a={0,1,2}
var b=a
var c=a
var d=a
var e=a
var f=a
var g=a
var h=a

1,a,b,c,d,e,f,g,h,2
0,1,2,0,0,0,0,0,0,1
0,1,2,2,0,0,0,0,0,1
0,1,2,2,2,0,0,0,0,1

The rule "Sticky" was proven Turing complete by ColorfulGabrielsp138. ColorfulGalaxy later discovered a "1 cell per generation" pattern in that rule today. It also has a special property that the population count is equal to the generation count starting from generation 12.

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#C Sticky 1CPG, with the additional feature that the tick number is equal to the population starting from generation 12
x = 14, y = 4, rule = Sticky
4A$3.A$A11bCB$4A!
[[ LABEL 0 9 8 "#I #P" ]]

[Citation needed]

ColorfulGalaxy found one in GUYTU6J's rule.

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#C 1CPG based on wickstretcher
x = 29, y = 96, rule = B3-jknr4ity5ijk6i8/S23-a4city6c7c
bo11b3o$4o9bo$o2bo10bob3o$bobo7b3obo3bo$b4o5b2ob2o4bo$b4ob3o2bo2bo2b2o
bo$7bo2bo3bo4b2o$5b2o4bob3o2b3o$8b3o8b2o$9bo7bobobo$11b2o3bo3b2o$13b2o
$12bo10bo$12b3o4b4obo$14b2o3bo4bo$15bob2o5bo$17b2o4bobo$18b2o2bobobo$
22bo3bo$17b2ob4o2bo$17bobo2bo$17b2o7bo$17b2o5b2obo$17b2o5bo$20bo$19bo
bo$20bo6$14b2o$b2o11bo2bo$2ob3o8bo4bo$bo2b2o6b2o$b4o6b2o2b4obo$7b2o2b
o5bo2bo$4bo2bo2bo2bob2o3b2o$4b3o2bo6bo3bobo$4b2o2b2o6bo2b2o$11b6o2bo$
12bo7bobo$13bo7b2o$13b2o6b3o$13bob2o3bo3bo$14b3o3bo4b2o$14b4o2b4ob2o$
15b2o8bo$19bo4bobo$19b3obobob2o$18bo8b2o$17b2o3bob2o$17bob2o2b2o$16b2o
bo4b2obo$27bo$20bo6bo$bo6bo10bobo$obo4bobo10bo$2obo3b2obo$bobo4bobo$b
ob2o3bob2o$2bobo4bobo$3bo6bo2$bo6bo$obo4bobo$2obo3b2obo$bobo4bobo$bob
2o3bob2o$2bobo4bobo$3bo6bo2$bo6bo$obo4bobo$2obo3b2obo$bobo4bobo$bob2o
3bob2o$2bobo4bobo$3bo6bo2$bo6bo$obo4bobo$2obo3b2obo$bobo4bobo$bob2o3b
ob2o$2bobo4bobo$3bo6bo2$bo6bo$obo4bobo$2obo3b2obo$bobo4bobo$bob2o3bob
2o$2bobo4bobo$3bo6bo!

Here is another multistate rule.

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#C B-Univ 1CPG (all non-zero cells counted)
x = 5, y = 19, rule = B-Univ
2AG$C.C$G2AE2$AGC$A.A$CGAE2$GCA$A.A$ACGE2$C2A$G.G$2ACG2$2AG$C.C$G3AE!

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#C B+Univ 1CPG with only 16 cells (all non-zero cells counted)
x = 6, y = 6, rule = BPlusUniv
2.A.A$.ACD.A$A$.AbDCA$2.A.A$3.A!

[b-engine]

...
Here is another multistate rule.

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#C B-Univ 1CPG (all non-zero cells counted)
x = 5, y = 19, rule = B-Univ
2AG$C.C$G2AE2$AGC$A.A$CGAE2$GCA$A.A$ACGE2$C2A$G.G$2ACG2$2AG$C.C$G3AE!

...

Smaller, at only initially 25 cells:

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x = 3, y = 14, rule = B-Univ
AG$ACE2$GC$2AE2$CA$GAE2$2A$CGE2$AG$ACG!
 

[Citation needed]

...
Here is another multistate rule.

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#C B-Univ 1CPG (all non-zero cells counted)
x = 5, y = 19, rule = B-Univ
2AG$C.C$G2AE2$AGC$A.A$CGAE2$GCA$A.A$ACGE2$C2A$G.G$2ACG2$2AG$C.C$G3AE!

...

Smaller, at only initially 25 cells:

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x = 3, y = 14, rule = B-Univ
AG$ACE2$GC$2AE2$CA$GAE2$2A$CGE2$AG$ACG!
 

Don't forget the patterns that you discovered.

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x rule = B-Univ
bCA$C$C$bC!

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x rule = B+Univ
E.CG!

[Citation needed]

ColorfulGalaxy found such a pattern that runs in a number of rules.

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#C Copyright rule 1CPG
#C The rule string was changed to avoid being flagged as advertising.
x = 42, y = 48, rule = 23-q8/32n
2o21b3o$obo20bo$o23bo$3bo22bo$4bo22b2o$3bobo21b2o$4b3o22b3o$5b3o22b2o
$9bo21b3o$7bo23b2o$9bobo2b2o18bo2b3o$11bo2bobo17bobo$12bobo24bo$13bo22b
o$13bo$15bo22b2o$14b3o21bobo$15bobo21bobo$16bo2bo21bo$19bo2$2b2o$bo2b
o$obo2bo$o2bobo$bo3b2o$2b3o$4bo4$2b2o$bo2bo$obo2bo$o2bobo$bo3b2o$2b3o
$4bo4$2b2o$bo2bo$obo2bo$o2bobo$bo3b2o$2b3o$4bo!

ColorfulGalaxy had been trying to complete one in rule "Codd" since the first day, but didn't actually complete it until today, when he spotted that oscillators with particular population signature can be built.

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#C Codd 1cpg
#C All non-zero cells are included in population count.
x = 28, y = 100, rule = Codd
4.5B$4.B.FAB$5BABAB$B.GABAG.A$BABA5B$BAF.F4A$9B$4.B2A.B$5BGBFB$B2A.B.
4A$BFBG5B$B.3A.F3A$9B4$7B$B4A.B$BA3BFB$BAB.BA19B$BAB.B19AB$BA3BA19B$B
5AB$7B$9.7B$9.B4A.B$10BA3BFB$B10AB.BA10B$10BAB.B10AB$9.BA3BA10B$9.B5A
B$9.7B$9.B4A.B$10BA3BFB$B10AB.BA9B$10BAB.B9AB$9.BA3BA9B$9.B5AB$8.8B$8.
B4A.B$9BA3BFB$B9AB.BA9B$9BAB.B9AB$8.BA3BA9B$8.B5AB$7.8B$7.B4A.B$8BA3B
FB$B8AB.BA9B$8BAB.B9AB$7.BA3BA9B$7.B5AB$6.8B$7B4A.B$B7A3BFB$7BAB.BA8B
$6.BAB.B8AB$6.BA3BA8B$6.B5AB$6.7B2$7.3B$7.BAB$4.4BA2B7.7B$5B4A.B4.4B4A
.B$B5A3BFB4.B4A3BFB$5BAB.BAB4.4BAB.BAB$4.BAB.BA4B4.BAB.BA4B$4.BA3B4AB
4.BA3B4AB$4.B5A4B4.B5A4B$4.3BA3B7.7B$6.BAB$6.3B2$2.7B$3B4A.B$B3A3BFB$
3BAB.BAB$2.BAB.BA4B$2.BA3B4AB$2.B5A4B$2.7B2$2.7B$3B4A.B$B3A3BFB$3BAB.
BAB$2.BAB.BA3B$2.BA3B3AB$2.B5A3B$2.7B2$2.7B$3B4A.B$B3A3BFB$3BAB.BAB$2.
BAB.BAB$2.BA3BAB$2.B5AB$2.7B!

[Citation needed]

Here is another rule discovered by one of ColorfulGalaxy's fellow countrymen from China.

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#C BasiKnight 1cpg
#C [[ ZOOM 4 THEME Catagolue ]]
x = 119, y = 19, rule = B2e3air4ijwz/S12-in3k4r5i
94bo20b3o$71b3o19b2o20bo$27bo20b3o20b3o18b2o$4b3o19b2o20bo22bo20b2o$4b
3o18b2o86b3o$4bo20b2o43bo18b4o19b4o$46b3o18b4o18b2obo18b3o2bo$3bo18b4o
19b4o40b4o22b2o$4o18b2obo18b3o2bo40bo$22b4o22b2o17b2o2b2o16b2o2b2o21b
2o$23bo44bo21bo23bo$2o2b2o16b2o2b2o17b2o2b2o17b2o2b2o16b2o2b2o17b2o2b
2o5$
2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2$b
2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2$b
!

[confocaloid]

Please avoid such multiposts, most of them break the forum rules. Edit the most recent post instead of multiposting.

Also, while the discussion topic itself (one-bit-per-tick growing patterns) is interesting, many of specific patterns you posted are fairly trivial examples, with little or no explanations of why these specific patterns deserve to be posted. There are lots and lots of similar patterns, and it does not make sense to post them all.

[Citation needed]

This pattern, found by ColorfulGalaxy, is in another replicator rule discovered by one of ColorfulGalaxy's fellow countrymen from China.

Code: Select all

#C Another one in islptng's rule
x = 10, y = 53, rule = BliptileExtCU
6A$A4.A.3A$A.2A.B2A.A$2A.2AC.3A$2.2A2$6A$A4.A.3A$A.2A.3A.A$2A.CBA.3A$
2.2A2$6A$A4.A.3A$A.CB.3A.A$2A.3A.3A$2.CB2$6A$A4.A.3A$A.2A.3A.A$CB.3A.
3A$2.2A2$6A$C4.A.3A$B.2A.3A.A$2A.3A.3A$2.2A2$BC4A$A4.A.3A$A.2A.3A.A$2A
.3A.3A$2.2A2$2ABC2A$A4.A.3A$A.2A.3A.A$2A.3A.3A$2.2A2$4ABC$A4.A.3A$A.2A
.3A.A$2A.3A.3A$2.2A2$6A$A4.B.3A$A.2A.C2A.A$2A.3A.3A$2.2A!

ColorfulGalaxy first noticed the arm extruder and the circuit triggering the growth.
Unfortunately, the growth was always "2 cells per odd number of generations".
Then he noticed the diode (it might have been an example of misnomer; it just looks like the diode from Wireworld) that can change the parity of the period, and he finally completed the pattern.
No extra population-adjusting oscillators were needed.

Caution: This rule is explosive.








EDIT: If you consider HistoricalLife, then a lone glider will work.

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x = 3, y = 3, rule = HistoricalLife
bo$BBA$3A!