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as p3, p13 and p17 had been discovered prior to 2000, trivial p39 and p51 oscillators could have been constructed
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A [[cellular automaton]] is said to be '''omniperiodic''' if it has [[oscillator]]s of all [[period]]s.
A [[cellular automaton]] is said to be '''omniperiodic''' if it has [[oscillator]]s of all [[period]]s.


[[Conway's Game of Life]] is omniperiodic, since there are oscillators of every period, with the last found period being {{period|41|text=p41}} in July {{year|2023}}. The last periods to be found are as follows:
[[Conway's Game of Life]] is omniperiodic, since there are oscillators of every period, with the last found period being {{period|41|text=p41}} in July {{year|2023}}. The last periods to be found, after stable Herschel conduits were found in {{year|1996}} that proved oscillators existed for all periods 61 and up, are as follows:


*{{period|39|text=p39}} (the first [[non-trivial]] example) in July {{year|2000}}
*[[p59 Herschel loop 1|p59]] in April {{year|1997}}
*{{period|27|text=p27}} in November {{year|2002}}
*[[54P17.1|p17]] and [[168P22.1|p22]] (the first [[non-trivial]] example of the latter) in April 1997
*{{period|51|text=p51}} (the first [[non-trivial]] example) in March {{year|2009}}
*[[258P3 on Achim's p11|p33]] (the first [[non-trivial]] example) in June 1997
*{{period|37|text=p37}} in April {{year|2009}}
*[[p57 Herschel loop 1|p57]] in December {{year|1997}}
*{{period|31|text=p31}} in November {{year|2010}}
*[[p49 bouncer loop|p49]] in August {{year|1999}}
*{{period|43|text=p43}} and {{period|53|text=p53}} in April {{year|2013}}
*[[134P39.1|p39]] (the first [[non-trivial]] example) in July {{year|2000}}
*{{period|23|text=p23}} in November {{year|2019}}
*[[123P27.1|p27]] in November {{year|2002}}
*{{period|38|text=p38}} in January {{year|2022}}
*[[Beluchenko's p51|p51]] (the first [[non-trivial]] example) in March {{year|2009}}
*{{period|34|text=p34}} (the first [[non-trivial]] example) in July {{year|2022}}
*[[Beluchenko's p37|p37]] in April {{year|2009}}
*{{period|19|text=p19}} and {{period|41|text=p41}} in July {{year|2023}}
*[[Merzenich's p31|p31]] in November {{year|2010}}
*[[p43 Snark loop|p43]] and [[P53 Snark loop|p53]] in April {{year|2013}}
*[[David Hilbert|p23]] in November {{year|2019}}
*[[Raucci's p38|p38]] in January {{year|2022}}
*[[74P34|p34]] (the first [[non-trivial]] example) in July {{year|2022}}
*[[Cribbage|p19]] and [[208P41|p41]] in July {{year|2023}}


==Large-period oscillators==
==Large-period oscillators==
Line 21: Line 26:


==Other rules==
==Other rules==
At least one oscillator of each period can be transferred from Life to [[Pedestrian Life]]<ref name="post163979" /> and [[Eightlife]]<ref name="post163994" /> to prove both rules omniperiodic. Additional proofs at least partially based on sets of Life oscillators have also been used to prove several isotropic non-totalistic rules close to Life omniperiodic.<ref name="post165056" /><ref name="post189573" /><ref name="post166947" />
By running [[Dean Hickerson]]'s [[drifter]] searcher to find signals that can be manipulated like Herschels, it is possible to prove that a variety of [[Life-like cellular automata]] are omniperiodic.<ref name="post2521" /><ref name="post139416" /> However, these signals are more like the [[2c/3 wire|2c/3 signal]], in that they operate on a dense background and can closely follow each other. There is a significant possibility that such a signal turner exists for Conway's Game of Life, but no explicit examples have been found.
By running [[Dean Hickerson]]'s [[drifter]] searcher to find signals that can be manipulated like Herschels, it is possible to prove that a variety of [[Life-like cellular automata]] are omniperiodic.<ref name="post2521" /><ref name="post139416" /> However, these signals are more like the [[2c/3 wire|2c/3 signal]], in that they operate on a dense background and can closely follow each other. There is a significant possibility that such a signal turner exists for Conway's Game of Life, but no explicit examples have been found.
Various isotropic non-totalistic rules besides those mentioned have also been shown to be omniperiodic through the discovery of simple spaceship reflectors. However, it has also been proven that not every rule that supports oscillators can be omniperiodic.<ref name="post164050" />


==See also==
==See also==
Line 29: Line 38:
==References==
==References==
<references>
<references>
<ref name="post163979">{{LinkForumThread
|format    = ref
|p          = 163979
|title      = Re: B38/S23
|author    = EvinZL
|date      = July 21, 2023
}}</ref>
<ref name="post163994">{{LinkForumThread
|format    = ref
|p          = 163994
|title      = Re: B3/S238
|author    = Naszvadi
|date      = July 21, 2023
}}</ref>
<ref name="post165056">{{LinkForumThread
|format    = ref
|p          = 165056
|title      = Re: Rules 1 transition from Life
|author    = Period1GliderGun
|date      = August 13, 2023
}}</ref>
<ref name="post189573">{{LinkForumThread
|format    = ref
|p          = 189573
|title      = Re: Rules 2 transitions from Life
|author    = WhiteHawk
|date      = July 11, 2024
}}</ref>
<ref name="post166947">{{LinkForumThread
|format    = ref
|p          = 166947
|title      = Re: Rules 1 transition from Life
|author    = Period1GliderGun
|date      = September 13, 2023
}}</ref>
<ref name="post2521">{{LinkForumThread
<ref name="post2521">{{LinkForumThread
|format    = ref
|format    = ref
Line 42: Line 86:
|author    = Dave Greene
|author    = Dave Greene
|date      = December 26, 2021
|date      = December 26, 2021
}}</ref>
<ref name="post164050 ">{{LinkForumThread
|format    = ref
|p          = 164050
|title      = Re: Unproven conjectures
|author    = toroidalet
|date      = July 22, 2023
}}</ref>
}}</ref>
</references>
</references>


==External links==
==External links==
* {{cite web
|url      = https://arxiv.org/abs/2312.02799
|title    = Conway's Game of Life is Omniperiodic
|author    = Nora Brown, Carson Cheng, Tanner Jacobi, Maia Karpovich, Matthias Merzenich, David Raucci, Mitchell Riley
|date      = December 5, 2023
|publisher = arXiv
}}
* {{LinkLexicon|lex_m.htm#myexperiencewithbheptominosinoscillators|patternname=My Experience with B-heptominos in Oscillators}}
* {{LinkLexicon|lex_m.htm#myexperiencewithbheptominosinoscillators|patternname=My Experience with B-heptominos in Oscillators}}
* {{LinkLexicon|lex_o.htm#omniperiodic}}
* {{LinkLexicon|lex_o.htm#omniperiodic}}

Latest revision as of 09:37, 5 September 2026

A cellular automaton is said to be omniperiodic if it has oscillators of all periods.

Conway's Game of Life is omniperiodic, since there are oscillators of every period, with the last found period being p41 in July 2023. The last periods to be found, after stable Herschel conduits were found in 1996 that proved oscillators existed for all periods 61 and up, are as follows:

Large-period oscillators

In October 1996, David Buckingham wrote the article My Experience with B-heptominos in Oscillators[1] that describes his discovery of Herschel conduits, including sufficient stable conduits to enable, for the first time, the construction of period n oscillators for every n ≥ 58, and true period n guns for every n ≥ 62.

The discovery of the Snark by Mike Playle in April 2013 allowed the construction of oscillators of all periods greater than or equal to 43.

Other rules

At least one oscillator of each period can be transferred from Life to Pedestrian Life[2] and Eightlife[3] to prove both rules omniperiodic. Additional proofs at least partially based on sets of Life oscillators have also been used to prove several isotropic non-totalistic rules close to Life omniperiodic.[4][5][6]

By running Dean Hickerson's drifter searcher to find signals that can be manipulated like Herschels, it is possible to prove that a variety of Life-like cellular automata are omniperiodic.[7][8] However, these signals are more like the 2c/3 signal, in that they operate on a dense background and can closely follow each other. There is a significant possibility that such a signal turner exists for Conway's Game of Life, but no explicit examples have been found.

Various isotropic non-totalistic rules besides those mentioned have also been shown to be omniperiodic through the discovery of simple spaceship reflectors. However, it has also been proven that not every rule that supports oscillators can be omniperiodic.[9]

See also

References

  1. ↑ Buckingham on B-heptomino/Herschel oscillators at Paul Callahan's Page of Conway's Life Miscellany
  2. ↑ EvinZL (July 21, 2023). Re: B38/S23 (discussion thread) at the ConwayLife.com forums
  3. ↑ Naszvadi (July 21, 2023). Re: B3/S238 (discussion thread) at the ConwayLife.com forums
  4. ↑ Period1GliderGun (August 13, 2023). Re: Rules 1 transition from Life (discussion thread) at the ConwayLife.com forums
  5. ↑ WhiteHawk (July 11, 2024). Re: Rules 2 transitions from Life (discussion thread) at the ConwayLife.com forums
  6. ↑ Period1GliderGun (September 13, 2023). Re: Rules 1 transition from Life (discussion thread) at the ConwayLife.com forums
  7. ↑ Matthias Merzenich (June 4, 2010). Re: B3/S234 (discussion thread) at the ConwayLife.com forums
  8. ↑ Dave Greene (December 26, 2021). Re: Suggested LifeWiki edits (discussion thread) at the ConwayLife.com forums
  9. ↑ toroidalet (July 22, 2023). Re: Unproven conjectures (discussion thread) at the ConwayLife.com forums


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