B36/S245
x=0, y = 0, rule = B36/S245
! #C [[ THEME Inverse ]]
#C [[ RANDOMIZE2 RANDSEED 1729 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]]
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LifeViewer -generated pseudorandom soup
Rulestring
245/36 B36/S245
Rule integer
26696
Character
Stable
Black/white reversal
B012578/S0134678
The rule sqrt replicator rule or B36/S245 is a Life-like cellular automaton in which cells survive from one generation to the next if they have 2, 4, or 5 neighbours and are born if they have 3 or 6 neighbours. Before the end of 2023, this rule was commonly referred to by the inaccurate name logarithmic replicator rule , due to its namesake object being misidentified.[1]
The cellular automaton has rulestring B36/S245, and differs from the "nearby" Move (B368/S245) in the lack of birth on 8 alive neighbours.
The time required to stabilize is generally much shorter than in Conway's Game of Life .
On August 19, 2020 , Peter Naszvadi constructed a Rule 110 unit cell in B36/S245, proving the rule Turing-complete .[2]
Notable patterns
The replicator
The name of this rule comes from an elementary replicator first discovered by Mark Niemiec . Unlike other replicators, (such as the one from HighLife ) this one does not reproduce itself cleanly , instead leaving oscillators behind which result in a more chaotic and slow growth pattern.[3]
x = 19, y = 3, rule = B36/S245
6o7b6o$o4bo7bo4bo$b4o9b4o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
#C [[ THUMBSIZE 2 ZOOM 10 ]]
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The namesake replicator with Θ(√t ) growth. It was initially assumed to have log(t) growth, leading to the incorrect and obsolete name.(click above to open LifeViewer ) RLE : here Plaintext : here
Its behaviour can be emulated by the following pattern, discovered by AforAmpere on April 11, 2020 in an isotropic non-totalistic rule:[4]
x = 3, y = 2, rule = B2a3eny4at/S01c2ci3i4e5e
bo$obo!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
#C [[ THUMBSIZE 2 ZOOM 10 ]]
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A small replicator that emulates the one above(click above to open LifeViewer )
This emulates the 4 -state elementary cellular automaton in a range-1 neighbourhood, with rule integer 508169473510261892743356039977344.
Growth bounds
Per the convention in Rule 120 , let b2 (n) be the number obtained from writing n in binary and reading this result as a base-4 number. (ie. 1110 = 10112 , so b2 (7) = 10114 = 6910 )[n 1]
In AforAmpere's emulator shown above,
the bottom edge increases to width 1 on generation 2 , then to width n on generation f(n) = 2*b2 (n-1)+b2 (⌊n-1 2 ⌋)+5 .
(lim infn→∞ (f(n) n2 ) = 3 4 , lim supn→∞ (f(n) n2 ) = 9 4 )
the top edge increases to width n on generation g(n) = 3*b2 (⌊n 2 ⌋+1)+6*b2 (⌊n+2-2⌊log2 (n+2)⌋ 2 ⌋)+2*(n mod 2)+5 .
(lim infn→∞ (g(n) n2 ) = 3 8 , lim supn→∞ (g(n) n2 ) = 3 4 )
In the t th iteration, the pattern's total height has the asymptotic bounds 2*√t < h(t) < 2*√5*t √3 .
The rule's original name stemmed from this asymptotic growth rate being interpreted as logarithmic in character, rather than square-root growth.
Spaceships
The rule has several known elementary spaceships, the smallest ones having speeds of c/4 orthogonal , 4c/23 orthogonal, and c/7 diagonal , shown below. Other known elementary spaceship speeds include c/2 orthogonal , c/3 orthogonal , c/5 orthogonal , 2c/5 orthogonal , c/6 orthogonal , c/7 orthogonal , 2c/7 orthogonal ,[5] c/3 diagonal, c/4 diagonal ,[6] c/5 diagonal ,[5] [7] c/6 diagonal ,[5] (2,1)c/5[5] and (2,1)c/6 .[8]
type
speed
period
first known
discoverer
minimal known
discoverer
elementary
c/2
2
17P2H1V0
unknown
4
51P4H2V0
unknown
26P4H2V0
DroneBetter, 2024
6
48P6H3V0
DroneBetter, 2024[5]
8
43P8H4V0
DroneBetter, 2024[5]
10
156P10H5V0
DroneBetter, 2024[5]
12
73P12H6V0
DroneBetter, 2024[5]
20
43P20H10V0
Jason Summers, 2002
52
86P52H26V0
Jason Summers, 2002
c/3
3
28P3H1V0
unknown
6
71P6H2V0
DroneBetter, 2024[5]
61P6H2V0
DroneBetter, 2024
9
154P9H3V0
DroneBetter, 2024[5]
c/4
4
13P4H1V0
unknown
c/5
5
246P5H1V0
unknown
69P5H1V0
JMM, 2024
2c/5
5
380P5H2V0
unknown
208P5H2V0
Darren Li , 2024[5]
c/6
6
32P6H1V0
unknown
c/7
7
44P7H1V0
Evan Clark, 2006
2c/7
7
504P7H2V0
DroneBetter, 2024[5]
14
68P14H4V0
DroneBetter, 2024[5]
4c/23
23
14P23H4V0
unknown
c/3d
3
60P3H1V1
unknown
c/4d
4
60P4H1V1
unknown
c/5d
5
146P5H1V1
DroneBetter, 2024[5]
c/6d
6
718P6H1V1
DroneBetter, 2024[5]
c/7d
7
Jellyfish (5P7H1V1)
Nathan Thompson , 1994
(2,1)c/5
5
544P5H2V1
DroneBetter, 2024[5]
(2,1)c/6
6
421P6H2V1
LaundryPizza03 , 2020[8]
corderoid
7c/150
300
58P300H14V0
Dean Hickerson , 1997
52P300H14V0
Benet Nasch, 2022[9]
7c/252
504
2715P504H14V0
FWKnightship,[10] Peter Naszvadi ,[11] toroidalet,[12] 2022
7c/501
2004
9980P2004H28V0
FWKnightship,[13] Peter Naszvadi,[14] 2022
7c/600
1200
280P1200H28V0
Dean Hickerson, 1997
x = 761, y = 182, rule = B36/S245
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o$270b6o117b4o4b4o78bob2o52b4o35bobo3b2o36bo3b3o20b2o7bo$270b2o2b2o117b5o2b5o80b2o90b2obo2bo37b2o3bo22b3o4bo$269b2o4b2o116b2o3b2o3b2o79b2o89bob2obo39b2obo32bo$268bob6obo116b3ob2ob3o79bob2o87bob2o3b2o36bo3bo24bo3bo2bobo$267bo3b4o3bo114bo2bob2obo2bo81bo87bobo2bo41b3o23bo2bob3o$266bo12bo113b2obo4bob2o80bobo85b2obobobo39bo2bo23bob2o2b2o$267bo10bo114b4o4b4o81b3o84bob3obo40b3o2b2o17bo3b2o$266bo12bo112b2obob4obob2o78bobo86bo2bobo42bo3bo19b2ob2obo$266bo12bo113bo10bo80bo3bo83b4o44bo2bo19b2obo$266bobo8bobo112bo2b8o2bo78bobo83b3obo43b3o3b2o17b2o3bo$265b2obo3b2o3bob2o111b2o10b2o79bo134b3o19bo3b2o$266b3o2b4o2b3o112bo12bo78bo84bo2b2o47b2o2b2o14bob2o2bo$265bob4ob2ob4obo111b2o3b4o3b2o78b2obo79b2obo2bo46b2o3b2o16b2o$266b14o114bob2o2b2obo81bo84bob2o47b4o13bob2obo$268bobo4bobo117bobo2bobo164bo3bo47b3obo14b6o$268bo8bo117bobo2bobo82b3o80b2o48bob2o16bobobo$267bob8obo206b2obo75b4obo48bo18b2obo$269bo2b2o2bo206bo4bo74b4o2b2o44bo4bo16b2o2bo$269bo2b2o2bo207b3o75bo7bo43b3ob2o15bobo$272b2o209b2obo76bo5bo43bob6o9b2o2bob2o$270b6o208bo78bo3bobo43bo2b2obo11b3obo$271bo2bo208b4o73bo6bo46b2o13bo2b2obo$270bob2obo206bob2ob2o72b2o4bo45b3o14bob3o$270b6o208b2o74b3o3b2o45bo2bo11b3o3bo$270bob2obo210b2o72b3o50b4o10b4o3bo$269bo6bo209b3o73b2o52bo8bobobo4bo$270b6o284bobo48bob4o7bo2b2o2bo$272b2o336bob2o2bo5b3o4bob2o$270bo4bo210bobo69bob2o47bo3b3o6b2ob2ob3obo$269bo2b2o2bo209b3o71b2o49bo3bo4b2ob2obobobo$270b6o208bo2bo70bob2o45bob5o5bo$269bo2b2o2bo206b5o71bo46bobobob2o4bo8b2o$270bob2obo207bo2b2o70b3o44bob2o2bo5b3o7bo$269b2o4b2o205b2ob3o117bo10bobo6bo$270b2o2b2o207b2o70bob2o45b2ob2o7b2o7bo$271bo2bo207bo3bo69bo46bobo3bo8bo6bo$270bo4bo207bo2bo69bo44b6o8b2o2bo4bo$486bo69bo43b2o2bobob2o5bo2bob2ob2o$486bo67bobo43b3obo2bo8bobo3bo$485bo68bo2bo41bobo6bo5b2obo3bo$553bo3bo42b2o10bobobo$553b3o45bo11b3o$554b2o46bo4b3obobo$553bo56b4o$550b2o53b2o3b3o$549bo2bo52bobobo$548bo3bo50b2obo4bo$547bobo2bo51bo7bo$547b3o51bobo7bo$550bo50bobo4b3o$546b2o2b2o48b2o6b2o$546bo3b2o51b2obob3o$545b3obo53bobo2bo$544b2o2b2o50bo4b2o$544bo3bo52bobo$545bo54bo$545b3o48bo2bo$546bo50b2o$544b2o$541b2obo49b3o$540bo2b2o45bo2b2o$541bo47bo2b3o$540b2o46b2o2b3o$588b2o4b2o$594bo$587bobo2bo$586bob2o2b2o$584bob5o2bo$584bo2b6o$587b2o$583b2obobo$578b2obo2bo$580bobobo$578bo2bobo$576bobo2bob2o$576b3o4b2o$578bobob3o$576bob3o$575bob5o$575b2ob4o$573bobobobo$572b2o$572b3o$573b2o$567b4obo$565bo2b2ob2o$564bo2b4o$565bobo2bo$563b2o2b2o2bo$565b4obo$564bo!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
#C [[ THUMBSIZE 2 WIDTH 3062.5 HEIGHT 742 ZOOM 4 ]]
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known elementary spaceships of minimal width and population for each speed, period and symmetry(click above to open LifeViewer )
In 1997 , Dean Hickerson discovered two replicator-based spaceships traveling at 7c/150 orthogonal and 7c/300 orthogonal respectively:
x = 38, y = 17, rule = B36/S245
15bobo$16bo4bobo$12bo3bo4bobo12bo$12bo2bobo2bo3bo9b2obo$12bo2bobo2bo3b
o9b2obo$12bo3bo4bobo12bo$16bo4bobo$15bobo4$b4o$3bo17bo$o2bo15b2obo$o2b
o15b2obo$3bo17bo$b4o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
#C [[ HEIGHT 500 WIDTH 600 THUMBSIZE 2 ZOOM 12 ]]
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14c/300o(click above to open LifeViewer ) RLE : here Plaintext : here Catagolue : here
x = 280, y = 163, rule = B36/S245
170bo$171bo$170bo16bo$127b3o38b2obo14bob2o$168b2obo14bob2o$170bo16bo$
171bo$170bo7$171bobo$170b2obo$169bo2bo15b2o$169b3o16bobo$169b3o16bobo$
169bo2bo15b2o$170b2obo$171bobo6$236bobo$170bo32bo15b2o14bo2b2o36b2o$
171bo31bo14bobo14bo3b2o34bobo$170bo16bo15bo14bobo14bo3b2o34bobo$168b2o
bo14bob2o29b2o14bo2b2o36b2o$127b3o38b2obo14bob2o46bobo$170bo16bo$171bo
$170bo6$237bo$94bo27bo27bo27bo41bo15bob2o37bo$93bob2o24bob2o24bob2o24b
ob2o38bob2o13b5o35bob2o$93bob2o24bob2o24bob2o24bob2o38bob2o13b5o35bob
2o$94bo27bo27bo27bo41bo15bob2o37bo$237bo4$4o24b4o24b4o48bo3b2o$81b2o
25bob2o2b2o$10b2o26b2o26b2o13b3o23b3o6b2o19b2o$9bobo25bobo25bobo38bobo
b2ob3o2b2o16bobo$9bobo25bobo25bobo38bobob2ob3o2b2o16bobo$10b2o26b2o26b
2o39b3o6b2o19b2o97bobo$108bob2o2b2o103b2o14bo2b2o36b2o$108bo3b2o89bo
14bobo14bo3b2o34bobo$203bo14bobo14bo3b2o34bobo$203bo15b2o14bo2b2o36b2o
$236bobo2$110bo$108b2obobo$20b2o86bo3b3o$19b4o84b2obo2bobobo$19b2o84b
2obo3bo2b4o18bo$20bo83bobo5bob4obo15b2obo$104bobo5bob4obo15b2obo$105b
2obo3bo2b4o18bo$85bo21b2obo2bobobo$85bo22bo3b3o$85bo22b2obobo$110bo15$
26bo$25bo59bo$26bobo56bo18bo$27bo57bo18bo$105b2o16bo13bo$102bob4o13b2o
bo10b2obo$102bob4o13b2obo10b2obo$105b2o16bo13bo$104bo$104bo4$241bo$
203bo15b2o12b3ob4ob2ob2o29b2o$203bo14bobo12bo3bo7bobo27bobo$203bo14bob
o12bo3bo7bobo27bobo$106bo11b2o99b2o12b3ob4ob2ob2o29b2o$10b2o26b2o26b2o
37b3o15b2o12b2o102bo$9bobo25bobo25bobo36b2o12b2o2bobo11bobo$9bobo25bob
o25bobo36b2o12b2o2bobo11bobo$10b2o26b2o26b2o13b3o21b3o15b2o12b2o$81b2o
23bo11b2o$4o24b4o24b4o3$234bo$235bo11bo$94bo27bo27bo27bo41bo14bo3bo4bo
bo30bo$93bob2o24bob2o24bob2o24bob2o38bob2o11b2o4b2o2bo3bo27bob2o$93bob
2o24bob2o24bob2o24bob2o38bob2o11b2o4b2o2bo3bo27bob2o$94bo27bo27bo27bo
41bo14bo3bo4bobo30bo$235bo11bo$234bo4$170bo2$163b2o6bob3o$161b2o7bo3bo
12bo$127b3o30bo3bo2b6obob2o8bob2o51bo$160bo3bo2b6obob2o8bob2o29b2o12b
3ob4ob2ob2o29b2o$161b2o7bo3bo12bo15bo14bobo12bo3bo7bobo27bobo$163b2o6b
ob3o27bo14bobo12bo3bo7bobo27bobo$203bo15b2o12b3ob4ob2ob2o29b2o$170bo
70bo5$169bo3b2o$169b3ob2obo$164bob2o4b3obo$164bo3b2o2b2ob3o10b2o$164b
3o4bobo14bobo$164b3o4bobo14bobo$164bo3b2o2b2ob3o10b2o$164bob2o4b3obo$
169b3ob2obo$169bo3b2o5$170bo2$163b2o6bob3o$161b2o7bo3bo12bo$160bo3bo2b
6obob2o8bob2o$127b3o30bo3bo2b6obob2o8bob2o$161b2o7bo3bo12bo$163b2o6bob
3o2$170bo!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
#C [[ HEIGHT 500 WIDTH 700 THUMBSIZE 2 ZOOM 2 ]]
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28c/1200o(click above to open LifeViewer ) RLE : here Plaintext : here
Linear growth
Replicators can also be used to create a gun for the c/7 diagonal ship:
x = 74, y = 45, rule = B36/S245
55b2o$55bobo$16bo38bobo$14b4o37bobo$13bo2bobo36bobo$13bo2bobo36b2o$14b
4o$16bo9$69b4o$68bo4bo$2b2o64b6o$bo2bo$bo2bo$6o$bo2bo28b6o$2b2o29bo4bo
$34b4o11$50b2o$49bobo$49bobo$49bobo$23bo25bobo$22b4o24b2o$21bobo2bo$
21bobo2bo$22b4o$23bo!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
#C [[ MAXGRIDSIZE 9 HEIGHT 600 WIDTH 800 THUMBSIZE 2 ZOOM 8 ]]
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(click above to open LifeViewer ) RLE : here Plaintext : here
On May 28, 2021, Luka Okanishi constructed the following period-408 4c/23 spaceship gun without synthesizing from smaller ships.[15]
x = 110, y = 64, rule = B36/S245
23b2obo2b2obo2bo38bo2bob2o2bob2o$23bob3obobo2b2o38b2o2bobob3obo$13bo2b
o6bob3obobo2b2o38b2o2bobob3obo6bo2bo$14b2obo5b2obo2b2obo2bo38bo2bob2o
2bob2o5bob2o$12b3obo76bob3o$15bo78bo12$29bo50bo$28b4o46b4o$27bobo2bo
44bo2bobo$5b3o19bobo2bo44bo2bobo19b3o$3bob2obo19b4o46b4o19bob2obo$5bo
23bo50bo23bo$4b3o47b2o47b3o$52bob2obo$52b2o2b2o$52bob2obo$54b2o$54b2o$
7b2obo88bob2o$5b4o5bo80bo5b4o$4bo2bo2b2ob2o80b2ob2o2bo2bo$4bo2bo2b2ob
2o80b2ob2o2bo2bo$5b4o5bo80bo5b4o$7b2obo88bob2o2$53b4o$54b2o$52bob2obo$
53bo2bo$54b2o$35b2o36b2o$35b2o16b4o16b2o$33bo4bo32bo4bo$2b4o28bo2bo34b
o2bo28b4o$2bo2bo26b2ob2ob2o13bo16b2ob2ob2o26bo2bo$2bo2bo27bob2obo13bob
o16bob2obo27bo2bo$2bo2bo45bobo50bo2bo$3b2o47b2o51b2o$52bobo$52bo5$2bo
2bo14bobo64bobo14bo2bo$2bo2bo14b2obo62bob2o14bo2bo$o2b2o2bo11bo4bo60bo
4bo11bo2b2o2bo$obo2bobo11bo4bo60bo4bo11bobo2bobo$20b2obo62bob2o$2b4o
14bobo12b2o36b2o12bobo14b4o$3b2o29bo2bo34bo2bo29b2o$35b2o36b2o$35b2o
36b2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
#C [[ MAXGRIDSIZE 9 HEIGHT 700 WIDTH 960 THUMBSIZE 2 ZOOM 8 ]]
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(click above to open LifeViewer )
On September 23, 2022, FWKnightship constructed a 14c/504 orthogonal replicator-based puffer ,[10] which Peter Naszvadi [11] and toroidalet [12] both found spaceship versions of on the following day. Also on September 24, FWKnightship found a second puffer with a speed of 28c/2004 orthogonal,[13] which Naszvadi completed into a spaceship two days later.[14]
Notes
See also
Rule 120 , another one-dimensional cellular automaton exhibiting Θ(√t ) growth.
References
↑ DroneBetter (November 22, 2023). Regarding the name of the """logarithmic""" replicator rule (discussion thread) at the ConwayLife.com forums
↑ Peter Naszvadi (August 19, 2020). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
↑ David Eppstein. "Replicators: B36/S245 ". Replicators . Retrieved on June 2, 2019.
↑ AforAmpere (April 11, 2020). Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums
↑ 5.00 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09 5.10 5.11 5.12 5.13 5.14 5.15 DroneBetter (March 3, 2024). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ B36/S245 at David Eppstein 's Glider Database
↑ May13 (March 3, 2024). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ 8.0 8.1 LaundryPizza03 (December 21, 2020). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ the attribution page , December 15, 2022
↑ 10.0 10.1 FWKnightship (September 23, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ 11.0 11.1 Peter Naszvadi (September 24, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ 12.0 12.1 toroidalet (September 24, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ 13.0 13.1 FWKnightship (September 24, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ 14.0 14.1 Naszvadi (September 26, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
↑ Luka Okanishi (May 28, 2021). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
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