Self-complementary - Revision history

Confocal: /* External links */ https://conwaylife.com/forums/viewtopic.php?p=170335#p170335

2023-11-02T18:41:52Z

External links: https://conwaylife.com/forums/viewtopic.php?p=170335#p170335

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	* {{LinkForumThread\|format=ref\|f=11\|t=3493\|title=Day and Knight rule group}}		* {{LinkForumThread\|format=ref\|f=11\|t=3493\|title=Day and Knight rule group}}
	* {{LinkForumThread\|format=ref\|f=11\|t=6125\|title=Hexagonal self-complementary rules}}		* {{LinkForumThread\|format=ref\|f=11\|t=6125\|title=Hexagonal self-complementary rules}}
			* {{LinkForumThread\|format=ref\|f=11\|t=6208\|title=The other five interesting self-complementary outer-totalistic rules}}

	[[Category:Self-complementary rules\| ]]		[[Category:Self-complementary rules\| ]]
	__NOTOC__		__NOTOC__

Confocal: links, reword, formatting

2023-10-09T04:16:25Z

links, reword, formatting

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	A two-state [[cellular automaton]] is said to be '''self-complementary''' if it remains unchanged under [[black/white reversal]], i.e. if a [[pattern]] composed of live [[cell]]s in a dead [[universe]] behaves the same as an equivalent pattern composed of dead cells in a live universe.		A two-state [[cellular automaton]] is said to be '''self-complementary''' if it remains unchanged under [[black/white reversal]], i.e. if a [[pattern]] composed of live [[cell]]s in a dead [[universe]] behaves the same as an equivalent pattern composed of dead cells in a live universe.

	A self-complementary rule's birth conditions completely determine ~~its~~ survival conditions, and vice versa, but no other constraints exist; as such there are 2<sup>9~~</sup> =~~ 512 distinct self-complementary [[Life-like cellular automaton\|Life-like rules]].		For a self-complementary rule, the birth conditions completely determine the survival conditions, and vice versa, but no other constraints exist; as such there are {{eq\|2{{sup\|9}}\|512}} distinct self-complementary [[Life-like cellular automaton\|Life-like rules]].

	Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal\|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.		Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal\|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.
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	==Examples==		==Examples==
	<!-- {{Main\|List of self-complementary rules}} --->		<!-- {{Main\|List of self-complementary rules}} --->
	The most well-known and well-investigated self-complementary rule is [[Day & Night]] (B3678/S34678); its equivalent strobing version is B01245/S0125.		The most well-known and well-investigated self-complementary rule is {{rl\|Day & Night}} (B3678/S34678); its equivalent strobing version is B01245/S0125.

	==Generalizations==		==Generalizations==

	Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still ~~lifes~~]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit<ref name="post138010" />. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.		Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still life]]s, [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit.<ref name="post138010" /> Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods.<ref name="post134179" />

	It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic rule\|cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day ~~and~~ Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.		It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic rule\|cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state.<ref name="post101012" /> Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like {{rl\|Day & Night}} by themselves, as well as when a pattern in one live state evolves on the background of the other live state.

	==References==		==References==
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	}}</ref>		}}</ref>
	</references>		</references>

			==External links==
			* {{LinkForumThread\|format=ref\|f=11\|t=3493\|title=Day and Knight rule group}}
			* {{LinkForumThread\|format=ref\|f=11\|t=6125\|title=Hexagonal self-complementary rules}}

	[[Category:Self-complementary rules\| ]]		[[Category:Self-complementary rules\| ]]
	__NOTOC__		__NOTOC__

Gzaytman: /* Generalizations */ The oscillator cyclic isn't relevant here.

2022-11-27T20:14:44Z

Generalizations: The oscillator cyclic isn't relevant here.

← Older revision		Revision as of 20:14, 27 November 2022
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	Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still lifes]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit<ref name="post138010" />. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.		Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still lifes]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit<ref name="post138010" />. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.

	It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.		It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic rule\|cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.

	==References==		==References==

Confocal: /* References */ cat

2022-11-27T19:56:08Z

References: cat

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	</references>		</references>

			[[Category:Self-complementary rules\| ]]
	__NOTOC__		__NOTOC__

Confocal at 19:38, 4 September 2022

2022-09-04T19:38:04Z

← Older revision		Revision as of 19:38, 4 September 2022
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	{{Glossary}}		{{Glossary}}
	A 2-state [[cellular ~~automata~~]] is said to be '''self-complementary''' if it remains unchanged under [[black/white reversal]], i.e. if a [[pattern]] composed of live [[cell]]s in a dead [[universe]] behaves the same as an equivalent pattern composed of dead cells in a live universe.		A two-state [[cellular automaton]] is said to be '''self-complementary''' if it remains unchanged under [[black/white reversal]], i.e. if a [[pattern]] composed of live [[cell]]s in a dead [[universe]] behaves the same as an equivalent pattern composed of dead cells in a live universe.

	A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 2<sup>9</sup> = 512 distinct self-complementary [[~~outer~~-~~totalistic~~ Life-like ~~cellular automata~~]].		A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 2<sup>9</sup> = 512 distinct self-complementary [[Life-like cellular automaton\|Life-like rules]].

	Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal\|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.		Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal\|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.

Book at 01:20, 10 December 2021

2021-12-10T01:20:36Z

← Older revision		Revision as of 01:20, 10 December 2021
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	==Generalizations==		==Generalizations==

	Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still ~~lives~~]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit<ref name="post138010" />. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.		Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still lifes]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit<ref name="post138010" />. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.

	It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.		It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.

Yoel at 06:19, 24 November 2021

2021-11-24T06:19:56Z

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	\|title = Re: Larger than Life		\|title = Re: Larger than Life
	\|p = 134179		\|p = 134179
	\|author = Yoel		\|author = Yoel Matveyev
	\|date = July 28, 2021		\|date = July 28, 2021
	}}</ref>		}}</ref>
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	\|title = Re: Gluonic		\|title = Re: Gluonic
	\|p = 101012		\|p = 101012
	\|author = Yoel		\|author = Yoel Matveyev
	\|date = July 24, 2020		\|date = July 24, 2020
	}}</ref>		}}</ref>
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	\|title = Re: Larger than Life		\|title = Re: Larger than Life
	\|p = 138010		\|p = 138010
	\|author = Yoel		\|author = Yoel Matveyev
	\|date = November 23, 2021		\|date = November 23, 2021
	}}</ref>		}}</ref>
	</references>		</references>
	__NOTOC__		__NOTOC__

Yoel at 06:17, 24 November 2021

2021-11-24T06:17:15Z

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	<ref name="post101012">{{LinkForumThread		<ref name="post101012">{{LinkForumThread
	\|format = ref		\|format = ref
	\|title = Re: ~~Larger than Life~~		\|title = Re: Gluonic
	\|p = 101012		\|p = 101012
	\|author = Yoel		\|author = Yoel

Yoel at 06:15, 24 November 2021

2021-11-24T06:15:16Z

← Older revision		Revision as of 06:15, 24 November 2021
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	==Generalizations==		==Generalizations==

	Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still lives]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.		Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still lives]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit<ref name="post138010" />. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.

	It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.		It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.
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	\|author = Yoel		\|author = Yoel
	\|date = July 24, 2020		\|date = July 24, 2020
			}}</ref>
			<ref name="post138010">{{LinkForumThread
			\|format = ref
			\|title = Re: Larger than Life
			\|p = 138010
			\|author = Yoel
			\|date = November 23, 2021
	}}</ref>		}}</ref>
	</references>		</references>
	__NOTOC__		__NOTOC__

GUYTU6J: red link removal

2021-11-06T08:35:29Z

red link removal

← Older revision		Revision as of 08:35, 6 November 2021
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	{{Glossary}}		{{Glossary}}
	~~An (outer~~-~~totalistic)~~ [[~~rule~~]] is said to be '''self-complementary''' if it remains unchanged under [[black/white reversal]], i.e. if a [[pattern]] composed of live [[cell]]s in a dead [[universe]] behaves the same as an equivalent pattern composed of dead cells in a live universe.		A 2-state [[cellular automata]] is said to be '''self-complementary''' if it remains unchanged under [[black/white reversal]], i.e. if a [[pattern]] composed of live [[cell]]s in a dead [[universe]] behaves the same as an equivalent pattern composed of dead cells in a live universe.

	A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 2<sup>9</sup> = 512 distinct self-complementary outer-totalistic ~~rules~~.		A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 2<sup>9</sup> = 512 distinct self-complementary [[outer-totalistic Life-like cellular automata]].

	Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal\|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.		Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal\|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.
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	==Generalizations==		==Generalizations==

	Self-complementary rules exhibiting complex behavior, [[spaceship~~\|spaceships~~]], [[still ~~life\|still lifes~~]], [[oscillator~~\|oscillators~~]], [[replicator~~\|replicators~~]], [[rake~~\|rakes~~]] and [[gun~~\|guns]] are known to exist in higher range [[Higher-range Outer-totalistic\|HROT~~]] ~~rules~~. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179"></~~ref~~>.		Self-complementary rules exhibiting complex behavior are known to exist in [[higher-range outer-totalistic]] rules, where [[spaceship]]s, [[still lives]], [[oscillator]]s, [[replicator]]s, [[rake]]s and [[gun]]s inhabit. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary [[1D cellular automata]] with higher ranges and custom neighborhoods<ref name="post134179" />.

	It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev~~\|Yoel Matveyev~~'s]] [[~~multistate~~ cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012"></~~ref~~>. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.		It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. [[Yoel Matveyev]]'s [[multistate rule\|multistate]] [[cyclic]] rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state<ref name="post101012" />. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule [[Rule:Triple-DN\|Triple-DN]] has 2 live states, which act exactly like [[Day and Night]] by themselves, as well as when a pattern in one live state evolves on the background of the other live state.

	==References==		==References==
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	}}</ref>		}}</ref>
	</references>		</references>
			__NOTOC__
	~~== External links ==~~
	~~{{LinkCatagolueRule\|r5b34t45s34t58}}~~

← Older revision		Revision as of 18:41, 2 November 2023
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	* {{LinkForumThread\|format=ref\|f=11\|t=3493\|title=Day and Knight rule group}}		* {{LinkForumThread\|format=ref\|f=11\|t=3493\|title=Day and Knight rule group}}
	* {{LinkForumThread\|format=ref\|f=11\|t=6125\|title=Hexagonal self-complementary rules}}		* {{LinkForumThread\|format=ref\|f=11\|t=6125\|title=Hexagonal self-complementary rules}}
			* {{LinkForumThread\|format=ref\|f=11\|t=6208\|title=The other five interesting self-complementary outer-totalistic rules}}

	[[Category:Self-complementary rules\| ]]		[[Category:Self-complementary rules\| ]]
	__NOTOC__		__NOTOC__