OCA:sqrt replicator rule

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B36/S245
x=64, y = 64, 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 ]]
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 ]]
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 ]]
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(11) = 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-12⌋)+5.
(lim infn→∞(f(n)n2) = 34, lim supn→∞(f(n)n2) = 94)
  • the top edge increases to width n on generation g(n) = 3*b2(⌊n2⌋+1)+6*b2(⌊n+2-2⌊log2(n+2)⌋2⌋)+2*(n mod 2)+5.
(lim infn→∞(g(n)n2) = 38, lim supn→∞(g(n)n2) = 34)

In the tth 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.

Oscillators

4P1
(Block)		 3P2
(Blinker)		 11P3 5P4.1 (rotating P-pentomino)
5P4.2 (rotating R-pentomino)
5P4.4 (rotating glider)		 20P5.1
20P5.2		 12P6 20P7 12P8 32P9 21P10 10P12 17P14

Replicator hasslers

40P119 27P127 44P204 48P408 48P612 22P816 48P1224 80P2040

Spaceships

The rule has known elementary spaceships of eighteen speeds (where those in bold have occurred naturally);

type direction speed period first known discoverer minimal known discoverer
elementary orthogonal c/2 2 17P2H1V0 Nathan Thompson, 1994
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 21P3H1V0 Saka, 2015[10]
6 71P6H2V0 DroneBetter, 2024[5] 57P6H2V0 DroneBetter, 2024
9 154P9H3V0 DroneBetter, 2024[5]
c/4 4 13P4H1V0 unknown
8 93P8H2V0 DroneBetter, 2024[5]
c/5 5 246P5H1V0 unknown 69P5H1V0 JMM, 2024
10 314P10H2V0 DroneBetter, 2024[5]
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 506P7H2V0 lordlouckster, 2022[11] 423P7H2V0 Darren Li, 2024[6]
14 68P14H4V0 DroneBetter, 2024[5]
3c/8 8 1372P8H3V0 Darren Li, 2024[6]
4c/23 23 14P23H4V0 unknown
diagonal c/3d 3 60P3H1V1 unknown
6 756P6H2V2 Yujh, 2024
c/4d 4 60P4H1V1 unknown 46P4H1V1 DroneBetter, 2024
c/5d 5 184P5H1V1 Josh Ball, 2015[12] 146P5H1V1 DroneBetter, 2024[5]
c/6d 6 718P6H1V1[n 2] DroneBetter, 2024[5] 64P6H1V1 Yujh, 2024[13]
c/7d 7 Jellyfish
(5P7H1V1)		 Nathan Thompson, 1994
c/8d 8 64P8H1V1 DroneBetter, 2024[6]
knightwise (2,1)c/5 5 544P5H2V1 DroneBetter, 2024[5]
(2,1)c/6 6 421P6H2V1 LaundryPizza03, 2020[9]
corderoid orthogonal 7c/150 300 58P300H14V0 Dean Hickerson, 1997 52P300H14V0 Benet Nasch, 2022[14]
7c/252 504 2715P504H14V0 FWKnightship,[15] Peter Naszvadi,[16] toroidalet,[17] 2022
7c/501 2004 9980P2004H28V0 FWKnightship,[18] Peter Naszvadi,[19] 2022
7c/600 1200 856P1200H28V0 Dean Hickerson, 1997
x = 1042, y = 265, rule = B36/S245 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#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 WIDTH 2660 HEIGHT 700 ZOOM 2.5 ]]
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 ]]
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 ]]
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 ]]
(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.[20]

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 ]]
(click above to open LifeViewer)

On September 23, 2022, FWKnightship constructed a 14c/504 orthogonal replicator-based puffer,[15] which Peter Naszvadi[16] and toroidalet[17] 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,[18] which Naszvadi completed into a spaceship two days later.[19]

Notes

  1. ↑ Sequence A000695 in the OEIS.
  2. ↑ 16 half-diagonals wide, 14 is disproven for a glide-symmetric so it is minimal
    is 718 cells, the smallest at width 16 is this one by yujh[13] is 351

See also

  • Rule 120, a one-dimensional cellular automaton exhibiting Θ(√t) growth.

References

  1. ↑ DroneBetter (November 22, 2023). Regarding the name of the """logarithmic""" replicator rule (discussion thread) at the ConwayLife.com forums
  2. ↑ Peter Naszvadi (August 19, 2020). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
  3. ↑ David Eppstein. "Replicators: B36/S245". Replicators. Retrieved on June 2, 2019.
  4. ↑ AforAmpere (April 11, 2020). Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums
  5. ↑ 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 5.16 DroneBetter (March 3, 2024). Re: B36/S245, in which the smallest known c/5d (by population) and (2,1)c/5 were found, together with minimal-widths: c/6d w16glide, 4c/14 w12e, 2c/7 w16e, 2c/5 w20e and (through Darren Li) w12a, all symmetries of c/2, c/3, c/4, c/5 and some period-multiples thereof
  6. ↑ 6.0 6.1 6.2 6.3 DroneBetter (April 27, 2024). Re: B36/S245, in which thinnest possible spaceships were found: c/6 w9a, 2c/7 w17o and (by Darren Li) w11a, 6c/12 w7a and w13o, w14e; 3c/8 (by Li again) w20e, and c/8d w12glide
  7. ↑ B36/S245 at David Eppstein's Glider Database
  8. ↑ May13 (March 3, 2024). Re: B36/S245, in which the smallest known c/5d (by bounding box) was found
  9. ↑ 9.0 9.1 LaundryPizza03 (December 21, 2020). Re: B36/S245, in which the (2,1)c/6 was found
  10. ↑ Saka (July 24, 2015). Re: Move/Morley (245/368), in which the smallest c/3 (polyglottal to both rules) was found
  11. ↑ lordlouckster (August 15, 2022). Message in #bots-and-mute-this on the Conwaylife Lounge Discord server, in which the first 2c/7 was found
  12. ↑ velcrorex (July 21, 2015). Re: Move/Morley (245/368), in which the shortest w15s c/5d at (which is the smallest by population of the thinnest possible family) was found, polyglottic to B36/S245
  13. ↑ 13.0 13.1 yujh (July 13, 2024). Re: B36/S245, in which the smallest known c/6d's of width 16 and 18 were found (with rlifesrc)
  14. ↑ the attribution page, December 15, 2022
  15. ↑ 15.0 15.1 15.2 15.3 FWKnightship (September 23, 2022). Re: B36/S245, showing a 14c/504 puffer
  16. ↑ 16.0 16.1 16.2 Peter Naszvadi (September 24, 2022). Re: B36/S245, corderising [15]'s puffer, then reducing it in an edit
  17. ↑ 17.0 17.1 toroidalet (September 24, 2022). Re: B36/S245, corderising [15]'s puffer more compactly than either of [16]'s
  18. ↑ 18.0 18.1 18.2 FWKnightship (September 25, 2022). Re: B36/S245, finding a 28c/2004 puffer
  19. ↑ 19.0 19.1 Naszvadi (September 26, 2022). Re: B36/S245, iin which [18]'s puffer was corderised
  20. ↑ Luka Okanishi (May 28, 2021). Re: B36/S245 (discussion thread) at the ConwayLife.com forums

External links

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