OCA:Sqrt replicator rule: Difference between revisions
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Neocaridina (talk | contribs) m Shrunk LifeViewer that was breaking the page |
DroneBetter (talk | contribs) m minpop reductions: -16 cells to 2c/6 (by searching the next glide-symmetric odd width up), -235 cells to 14c/1200 corderoid (by replacing 3-engine egg puffer with natural 2-engine one) |
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|rowspan="3"|c/3||3||{{LinkCatagolue|xq3_6jrrj6z6h77h6|b36s245|name=28P3H1V0|style=raw}}||unknown||style="background: #80ff80;"|{{LinkCatagolue|xq3_1v33fmr33|b36s245|name=21P3H1V0|style=raw}}||Saka, 2015<ref name="post21408" /> | |rowspan="3"|c/3||3||{{LinkCatagolue|xq3_6jrrj6z6h77h6|b36s245|name=28P3H1V0|style=raw}}||unknown||style="background: #80ff80;"|{{LinkCatagolue|xq3_1v33fmr33|b36s245|name=21P3H1V0|style=raw}}||Saka, 2015<ref name="post21408" /> | ||
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|6||{{LinkCatagolue|xq6_06jrrj6z06h77h6zgkukga8z0fsdiz04sg1z01okez6a7|b36s245|name=71P6H2V0|style=raw}}||DroneBetter, 2024<ref name="post179555" />||{{LinkCatagolue|xq6_a6jei0iej6azx9k4k9zx4ne29z0615010f64|b36s245|name=57P6H2V0|style=raw}}||DroneBetter, 2024 | |6||{{LinkCatagolue|xq6_06jrrj6z06h77h6zgkukga8z0fsdiz04sg1z01okez6a7|b36s245|name=71P6H2V0|style=raw}}||DroneBetter, 2024<ref name="post179555" />||<!--{{LinkCatagolue|xq6_a6jei0iej6azx9k4k9zx4ne29z0615010f64|b36s245|name=57P6H2V0|style=raw}}||DroneBetter, 2024-->{{LinkCatagolue|xq6_17rjl8wdjb71zx3b36d3|b36s245|name=41P6H2V0|style=raw}}||DroneBetter, 2026 | ||
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name="post151314" /> [[Peter Naszvadi]],{{refn|name=post151339|{{LinkForumThread|p=151339|title=Re: B36/S245|author=Peter Naszvadi|date=September 24, 2022|format=raw}}, corderising <ref name="post151314" />'s puffer, then reducing it in an edit}} toroidalet,{{refn|name=post151345|{{LinkForumThread|p=151345|title=Re: B36/S245|author=toroidalet|date=September 24, 2022|format=raw}}, corderising <ref name="post151314" />'s puffer more compactly than either of <ref name="post151339" />'s}} 2022 | |7c/252||504|| || 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name="post151314" /> [[Peter Naszvadi]],{{refn|name=post151339|{{LinkForumThread|p=151339|title=Re: B36/S245|author=Peter Naszvadi|date=September 24, 2022|format=raw}}, corderising <ref name="post151314" />'s puffer, then reducing it in an edit}} toroidalet,{{refn|name=post151345|{{LinkForumThread|p=151345|title=Re: B36/S245|author=toroidalet|date=September 24, 2022|format=raw}}, corderising <ref name="post151314" />'s puffer more compactly than either of <ref name="post151339" />'s}} 2022 | ||
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|7c/501||2004|| || ||9980P2004H28V0<!--lifelib will not return apgcode when running overnight-->||FWKnightship,<ref name="post151347" /> Peter Naszvadi,{{refn|name=post151388|{{LinkForumThread|f=11|t=1887|p=151388|title=Re: B36/S245|author=Naszvadi|date=September 26, 2022|format=raw}}, | |7c/501||2004|| || ||9980P2004H28V0<!--lifelib will not return apgcode when running overnight-->||FWKnightship,<ref name="post151347" /> Peter Naszvadi,{{refn|name=post151388|{{LinkForumThread|f=11|t=1887|p=151388|title=Re: B36/S245|author=Naszvadi|date=September 26, 2022|format=raw}}, in which <ref name="post151347" />'s puffer was corderised}} 2022 | ||
|- | |- | ||
|7c/600||1200 | |7c/600||1200||{{LinkCatagolue|xq1200_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|b36s245|name=856P1200H28V0|style=raw}}||Dean Hickerson, 1997||{{LinkCatagolue|xq1200_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|b36s245|name=621P1200H28V0|style=raw}}||DroneBetter, 2026<ref>{{CiteDiscord|channel=naturalistic|id=1511049351118717039|author=DroneBetter|date=June 1, 2026}}; done by retrofitting a {{LinkCatagolue|yl1200_1_8_ae52b54e872a6c328224e18b1ce646b3|b36s245|name=2-engine egg-puffer from a soup|style=raw}} in place of the 3-engine one used four times in Hickerson's corderoid</ref> | ||
|} | |} | ||
{{EmbedViewer | {{EmbedViewer | ||
|rle = x = 1042, y = 265, rule = B36/S245 | |rle = x = 1042, y = 265, rule = B36/S245 | ||
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|viewerconfig = #C [[ THUMBSIZE 3 WIDTH 2660 HEIGHT 700 ZOOM 2.5 ]] | |viewerconfig = #C [[ THUMBSIZE 3 WIDTH 2660 HEIGHT 700 ZOOM 2.5 ]] | ||
|caption = known elementary spaceships of minimal width and population for each speed, period and symmetry | |caption = known elementary spaceships of minimal width and population for each speed, period and symmetry | ||
Revision as of 18:06, 1 June 2026
| B36/S245 | |
| Rulestring | 245/36 B36/S245 |
|---|---|
| Rule integer | 26696 |
| Character | Stable |
| Black/white reversal | B012578/S0134678 |
The 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]
| 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]
| 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
Replicator hasslers
| 64P50 | 40P119 | 27P127 | 22P204 | 48P408 | 22P612 | 30P804 | 22P816 | 48P1224 | 80P2040 |
Spaceships
The rule has known elementary spaceships of eighteen speeds (where those in bold have occurred naturally, and italic seminaturally);
- orthogonal: c/2, c/3, c/4, c/5, 2c/5, c/6, c/7, 2c/7,[5] 3c/8,[6] 4c/23.
- diagonal: c/3, c/4,[7] c/5,[5][8] c/6,[5] c/7, c/8
- knightwise: (2,1)c/5[5] and (2,1)c/6.[9]
Green means a spaceship has been proven to have minimal population via LLSSS.
| 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] | 41P6H2V0 | DroneBetter, 2026 | |||
| 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 | 158P5H2V0 | DroneBetter, 2025 | ||
| 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 | 169P6H2V2 | DroneBetter, 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] | 61P6H1V1[n 3] | DroneBetter, 2025 | ||
| 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 | 621P1200H28V0 | DroneBetter, 2026[20] |
| 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:
| 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:
| (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.[21]
| (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
- ↑ Sequence A000695 in the OEIS.
- ↑ 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 - ↑ note that although this table uses minpop, it is not minimal by all metrics; its averagepop is 75+13, compared to 68 for the previous entry, 64P6H1V1 by Yujh, 2024[13] (which also have a maxpop less than this one's averagepop)
See also
- Rule 120, a 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 , 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
- ↑ 6.0 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
- ↑ B36/S245 at David Eppstein's Glider Database
- ↑ May13 (March 3, 2024). Re: B36/S245 , in which the smallest known c/5d (by bounding box) was found
- ↑ 9.0 9.1 LaundryPizza03 (December 21, 2020). Re: B36/S245 , in which the (2,1)c/6 was found
- ↑ Saka (July 24, 2015). Re: Move/Morley (245/368) , in which the smallest c/3 (polyglottal to both rules) was found
- ↑ lordlouckster (August 15, 2022). Message in #bots-and-mute-this on the Conwaylife Lounge Discord server , in which the first 2c/7 was found
- ↑ 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
- ↑ 13.0 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)
- ↑ the attribution page, December 15, 2022
- ↑ 15.0 15.1 15.2 15.3 FWKnightship (September 23, 2022). Re: B36/S245 , showing a 14c/504 puffer
- ↑ 16.0 16.1 16.2 Peter Naszvadi (September 24, 2022). Re: B36/S245 , corderising [15]'s puffer, then reducing it in an edit
- ↑ 17.0 17.1 toroidalet (September 24, 2022). Re: B36/S245 , corderising [15]'s puffer more compactly than either of [16]'s
- ↑ 18.0 18.1 18.2 FWKnightship (September 25, 2022). Re: B36/S245 , finding a 28c/2004 puffer
- ↑ 19.0 19.1 Naszvadi (September 26, 2022). Re: B36/S245 , in which [18]'s puffer was corderised
- ↑ DroneBetter (June 1, 2026). Message in #naturalistic on the Conwaylife Lounge Discord server ; done by retrofitting a 2-engine egg-puffer from a soup in place of the 3-engine one used four times in Hickerson's corderoid
- ↑ Luka Okanishi (May 28, 2021). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
External links
- sqrt replicator rule at Adam P. Goucher's Catagolue
- B36/S245 at David Eppstein's Glider Database
- B36/S245 (discussion thread) at the ConwayLife.com forums