Difference between revisions of "OCA:Sqrt replicator rule"
Haycat2009 (talk | contribs) m |
(I added new proposed move, since the issue is open (there's no convincing explanation for why the existing name in use should be removed from the page, and a new name not in common use should be added instead). Restore Revision as of 01:50, 9 December 2023, which mentions both the common name and the existing disagreement on this issue) Tag: Manual revert |
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|ruleinteger = 26696 | |ruleinteger = 26696 | ||
|reversal = B012578/S0134678 | |reversal = B012578/S0134678 | ||
}} | }}{{move|1=OCA:B36/S245|2=The new name was not used in actual discussions about the rule or objects in it, and is a modification (replacement of a single word) of the previous name 'logarithmic replicator rule'; the latter is in use and is recognizable.<br />Discussion thread: https://conwaylife.com/forums/viewtopic.php?f=16&t=6240}} | ||
The rule '''Sqrt replicator rule''', | The rule '''B36/S245''' ([[Catagolue]] name '''Sqrt replicator rule'''; however, the rule is commonly known by the name '''logarithmic replicator rule''') is a [[Life-like cellular automaton]] in which [[cell]]s survive from one [[generation]] to the next if they have 2, 4, or 5 [[Moore neighbourhood|neighbours]] and are born if they have 3 or 6 neighbours. | ||
The cellular automaton has [[rulestring]] B36/S245, and differs from | The cellular automaton has [[rulestring]] B36/S245, and differs from a "nearby" {{rl|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]]. | The time required to stabilize is generally much shorter than in [[Conway's Game of Life]]. | ||
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===The replicator=== | ===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 {{rl|HighLife}}) this one does not reproduce itself [[clean]]ly, instead leaving oscillators behind which result in a more chaotic | The name of this rule comes from an elementary [[replicator]] first discovered by [[Mark Niemiec]]. Unlike other replicators, (such as the one from {{rl|HighLife}}) this one does not reproduce itself [[clean]]ly, instead leaving oscillators behind which result in a more chaotic growth pattern.<ref>{{cite web|url=https://www.ics.uci.edu/~eppstein/ca/replicators/b36s245.html|title=Replicators: B36/S245|author=David Eppstein|work=Replicators|accessdate=June 2, 2019}}</ref> | ||
{{EmbedViewer | {{EmbedViewer | ||
Revision as of 06:49, 9 December 2023
| B36/S245 | |
| Rulestring | 245/36 B36/S245 |
|---|---|
| Rule integer | 26696 |
| Character | Stable |
| Black/white reversal | B012578/S0134678 |
|
It has been proposed that this page be moved to OCA:B36/S245 due to the following reason: The new name was not used in actual discussions about the rule or objects in it, and is a modification (replacement of a single word) of the previous name 'logarithmic replicator rule'; the latter is in use and is recognizable. Discussion thread: https://conwaylife.com/forums/viewtopic.php?f=16&t=6240 |
The rule B36/S245 (Catagolue name Sqrt replicator rule; however, the rule is commonly known by the name logarithmic replicator rule) 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. The cellular automaton has rulestring B36/S245, and differs from a "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.[1]
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 growth pattern.[2]
| The namesake replicator with Θ(√t) growth. (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:[3]
| 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
|
This article may require cleanup to meet LifeWiki's quality standards. |
Per the convention in Rule 225, 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-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.
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, c/3 diagonal, c/4 diagonal,[4] and (2,1)c/6.[5]
| (click above to open LifeViewer) RLE: here Plaintext: here |
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.[6]
| (click above to open LifeViewer) |
On September 23, 2022, FWKnightship constructed a 14c/504 orthogonal replicator-based puffer,[7] which Peter Naszvadi[8] and toroidalet[9] 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.[10]
Notes
See also
- Rule 225, another one-dimensional cellular automaton exhibiting Θ(√t) growth.
References
- ↑ 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
- ↑ B36/S245 at David Eppstein's Glider Database
- ↑ LaundryPizza03 (December 21, 2020). 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
- ↑ FWKnightship (September 23, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
- ↑ Peter Naszvadi (September 24, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
- ↑ toroidalet (September 24, 2022). Re: B36/S245 (discussion thread) at the ConwayLife.com forums
- ↑ FWKnightship (September 24, 2022). 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