Difference between revisions of "OCA:2×2"
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'''2×2''' is a [[ | '''2×2''' is a [[Life-like cellular automaton]] in which [[cell]]s survive from one generation to the next if they have 1, 2 or 5 [[neighbours]], and are born if they have 3 or 6 neighbours. It thus has rulestring "B36/S125". Patterns under the rule have a chaotic evolution similar to those under the standard [[Life]], but the chaos tends to die out much more quickly. | ||
Its name comes from the fact that patterns made up of 2 | Its name comes from the fact that [[2×2 block rule|patterns made up of {{times|2|2}} blocks]] continue to evolve as patterns made up of {{times|2|2}} blocks. | ||
==Block evolution== | ==Block evolution== | ||
The 2×2 rule can emulate a simpler cellular automaton that acts on each 2 | The 2×2 rule can emulate a simpler cellular automaton that acts on each {{times|2|2}} block. The emulated automaton is a [[block cellular automaton]] that makes use of the [[Margolus neighbourhood]] and evolves according to the following six rules: | ||
[[ | [[File:2x2block evolve.png|thumb|center|The 2×2 block evolution rule]] | ||
Note that, as this emulates a Margolus neighbourhood, the resulting block appears at the center of the original four blocks. Thus, patterns that are originally made up of 2 | Note that, as this emulates a Margolus neighbourhood, the resulting block appears at the center of the original four blocks. Thus, patterns that are originally made up of {{times|2|2}} blocks will forever be made up of {{times|2|2}} blocks, but the block partition will be offset by one [[cell]] in the odd [[generation]]s from the even generations. By examining the image above, one can see that an isotropic cellular automaton will emulate a Margolus block cellular automaton if and only if the following four equations are satisfied: {{eq|B4w|S4q}}, {{eq|B5a|S6a|S7c}}, {{eq|B3i|S5i}}, {{eq|B1c|B2a|S3a}}, where the first equation for example means that the birth condition for cells with four neighbours must equal the survival condition for cells with four neighbours. There are {{eq|2<sup>12</sup>|4096}} such rules, which emulate {{eq|2<sup>6</sup>|64}} different block cellular automata. Any arrangement of cells that fits within a {{times|2|2}} bounding box can simulate these using isotropic non-totalistic rules. | ||
This rule can be seen to satisfy the above equations because 4 is neither a birth condition nor a survival condition, 5 is not a birth condition and 6 and 7 are not survival conditions, 3 is a birth condition and 5 is a survival condition, and 3 is not a survival condition and 1 and 2 are not birth conditions. | This rule can be seen to satisfy the above equations because 4 is neither a birth condition nor a survival condition, 5 is not a birth condition and 6 and 7 are not survival conditions, 3 is a birth condition and 5 is a survival condition, and 3 is not a survival condition and 1 and 2 are not birth conditions. | ||
The [[non-totalistic | The [[non-totalistic cellular automaton]] B3i4int5ey6k7e/S1e2k3ey4irt5i can be used to simulate this rule. {{times|1|1}} cells simulate the clusters of {{times|2|2}} blocks, and only every second generation plays, since odd generations have the offset. Since this rule is self-similar when scaling up (patterns made of {{times|4|4}} blocks will remain made of {{times|4|4}} blocks every second generation, {{times|8|8}} every fourth, etc.) this rule itself can simulate the mentioned {{times|2|2}} oscillators. | ||
==Notable patterns== | ==Notable patterns== | ||
A large variety of [[still life]]s and [[oscillators]] appear spontaneously from randomly generated starting states. There is also a somewhat rare naturally-occurring [[spaceship]], which travels at c/8 diagonally. | A large variety of [[still life]]s and [[oscillators]] appear spontaneously from randomly generated starting states. There is also a somewhat rare naturally-occurring [[spaceship]], which travels at c/8 diagonally. | ||
===Still | ===Still lives=== | ||
Still | Still lives are generally smaller in 2×2 than in Life, with the smallest occurring having a population of just 2 cells. These still life patterns still tend to be similar to Life patterns in terms of structure, for example often having islands that stabilise each other. Many still lives from Life are also still lives in 2×2, for example, the [[beehive]], [[tub]], [[loaf]], [[pond]] and [[mango]]. | ||
population of just 2 cells. These still life patterns still tend to be similar to Life patterns in terms | |||
of structure, for example often having islands that stabilise each other. | |||
Many still | |||
{{EmbedViewer | {{EmbedViewer | ||
|pname = 2x2stills | |pname = 2x2stills | ||
|viewerconfig = #C [[ THUMBSIZE 2 ]] | |viewerconfig = #C [[ THUMBSIZE 2 ]] | ||
|position = center | |||
|caption = Some sample still lifes | |||
|style = width:300px; | |||
}} | }} | ||
====Enumerating still | ====Enumerating still lives==== | ||
The following table catalogs all still | The following table catalogs all still lives in the 2×2 rule with 10 or fewer [[cell]]s.<ref>Computed using the {{filename|EnumStillLifes.c}} script located [https://conwaylife.com/forums/viewtopic.php?f=9&t=44 here].</ref> | ||
{| class="wikitable" style="margin-left:auto;margin-right:auto;" | {| class="wikitable" style="margin-left:auto;margin-right:auto;" | ||
Line 101: | Line 99: | ||
|} | |} | ||
===Common still | ===Common still lives=== | ||
The following table lists the twenty most common [[strict still life]]s that arise after several generations of a [[soup|random starting pattern]].<ref name="freqres">Full results are located [ | The following table lists the twenty most common [[strict still life]]s that arise after several generations of a [[soup|random starting pattern]].<ref name="freqres">Full results are located [https://conwaylife.com/forums/download/file.php?id=18 here].</ref> The "approx. rel. freq." column gives an estimate of the proportion of all randomly-occurring still lives that will be of the given type. | ||
<table border="0" cellpadding="0" cellspacing="0" width="100%"><tr><td align="center" width="50%" valign="top"> | <table border="0" cellpadding="0" cellspacing="0" width="100%"><tr><td align="center" width="50%" valign="top"> | ||
{| class="wikitable" style="margin-left:auto;margin-right:auto;" | {| class="wikitable" style="margin-left:auto;margin-right:auto;" | ||
Line 117: | Line 115: | ||
|- | |- | ||
! 2 | ! 2 | ||
| [[Image:2x2_stilllife_rank2.png]] | | [[Image:2x2_stilllife_rank2.png]] ([[duoplet]]) | ||
| 2 | | 2 | ||
| 0.251 | | 0.251 | ||
Line 177: | Line 175: | ||
| [[Image:2x2_stilllife_rank12.png]] ([[loaf]]) | | [[Image:2x2_stilllife_rank12.png]] ([[loaf]]) | ||
| 7 | | 7 | ||
| 5.8 | | {{times|5.8|10<sup>-4</sup>}} | ||
|- | |- | ||
! 13 | ! 13 | ||
| [[Image:2x2_stilllife_rank13.png]] | | [[Image:2x2_stilllife_rank13.png]] | ||
| 6 | | 6 | ||
| 5.63 | | {{times|5.63|10<sup>-4</sup>}} | ||
|- | |- | ||
! 14 | ! 14 | ||
| [[Image:2x2_stilllife_rank14.png]] | | [[Image:2x2_stilllife_rank14.png]] | ||
| 6 | | 6 | ||
| 4.04 | | {{times|4.04|10<sup>-4</sup>}} | ||
|- | |- | ||
! 15 | ! 15 | ||
| [[Image:2x2_stilllife_rank15.png]] | | [[Image:2x2_stilllife_rank15.png]] | ||
| 7 | | 7 | ||
| 2.56 | | {{times|2.56|10<sup>-4</sup>}} | ||
|- | |- | ||
! 16 | ! 16 | ||
| [[Image:2x2_stilllife_rank16.png]] ([[aircraft carrier]]) | | [[Image:2x2_stilllife_rank16.png]] ([[aircraft carrier]]) | ||
| 6 | | 6 | ||
| 2.23 | | {{times|2.23|10<sup>-4</sup>}} | ||
|- | |- | ||
! 17 | ! 17 | ||
| [[Image:2x2_stilllife_rank17.png]] ([[pond]]) | | [[Image:2x2_stilllife_rank17.png]] ([[pond]]) | ||
| 8 | | 8 | ||
| 1.94 | | {{times|1.94|10<sup>-4</sup>}} | ||
|- | |- | ||
! 18 | ! 18 | ||
| [[Image:2x2_stilllife_rank18.png]] ([[mango]]) | | [[Image:2x2_stilllife_rank18.png]] ([[mango]]) | ||
| 8 | | 8 | ||
| 1.28 | | {{times|1.28|10<sup>-4</sup>}} | ||
|- | |- | ||
! 19 | ! 19 | ||
| [[Image:2x2_stilllife_rank19.png]] | | [[Image:2x2_stilllife_rank19.png]] | ||
| 5 | | 5 | ||
| 9.6 | | {{times|9.6|10<sup>-5</sup>}} | ||
|- | |- | ||
! 20 | ! 20 | ||
| [[Image:2x2_stilllife_rank20.png]] | | [[Image:2x2_stilllife_rank20.png]] | ||
| 6 | | 6 | ||
| 7.68 | | {{times|7.68|10<sup>-5</sup>}} | ||
|} | |} | ||
</td></tr></table> | </td></tr></table> | ||
Line 229: | Line 227: | ||
{{EmbedViewer | {{EmbedViewer | ||
|pname = 2x2period2oscillators | |pname = 2x2period2oscillators | ||
|viewerconfig = #C [[ THUMBSIZE 2 AUTOSTART GPS 2 ]] | |viewerconfig = #C [[ THUMBSIZE 2 AUTOSTART GPS 2 ]] | ||
|position = center | |||
|caption = Some period 2 oscillators | |||
|style = width:300px; | |||
}} | }} | ||
====Higher-period oscillators==== | ====Higher-period oscillators==== | ||
One of the most interesting aspects of the 2×2 rule is the large number of naturally-occurring higher-period oscillators. Oscillators with periods 3, 4, 5, 6, 10, 14, 22 and 26 are all relatively frequent, and oscillators are also known for periods 8, 11, 12, 17, 24 and | One of the most interesting aspects of the 2×2 rule is the large number of naturally-occurring higher-period oscillators. Oscillators with periods 3, 4, 5, 6, 10, 14, 22 and 26 are all relatively frequent, and oscillators are also known for periods 7, 8, 11, 12, 17, 20, 24, 28, 30, 60, 62 and 126. | ||
{{EmbedViewer | {{EmbedViewer | ||
|pname = | |pname = 2x2oscillatorsnew | ||
|viewerconfig = #C [[ THUMBSIZE 2 HEIGHT 720 WIDTH 960 ]] | |viewerconfig = #C [[ THUMBSIZE 2 HEIGHT 720 WIDTH 960 ]] | ||
|position = center | |||
|caption = A [[stamp collection]] of oscillators with different periods from 2 through 126 | |||
|style = width:300px; | |||
}} | }} | ||
One simple infinite family of oscillators is given by the 2 | One simple infinite family of oscillators is given by the {{times|2|(4n)}} boxes of alive [[cell]]s.<ref>{{cite web|url=http://www.nathanieljohnston.com/index.php/2009/05/rectangular-oscillators-in-the-2x2-b36s125-cellular-automaton/|title=Rectangular Oscillators in the 2×2 (B36/S125) Cellular Automaton|author=Nathaniel Johnston|date=May 22, 2009|accessdate=May 24, 2009}}</ref> Such oscillators can be analyzed by noting that each [[phase]] of their oscillation can be represented as an exclusive or (XOR) of rectangles of different sizes that emulate the [[Rule 90]] cellular automaton.<ref>{{cite web|url=https://groups.google.com/group/comp.theory.cell-automata/browse_frm/thread/b059a5fb0c796743/c5683380affb8068?lnk=gst&q=2x2#c5683380affb8068|title=Life 2x2: long oscillator|publisher=comp.theory.cell-automata|accessdate=May 24, 2009|date=November 2, 2001}}</ref> The period of these oscillators for n = 1, 2, 3, ... is given by the sequence 2, 6, 14, 14, 62, 126, 30, 30, 1022, ... (Sloane's {{OEIS|A160657}}). | ||
====Naturally occurring oscillators==== | ====Naturally occurring oscillators==== | ||
Line 390: | Line 390: | ||
{{EmbedViewer | {{EmbedViewer | ||
|pname = 2x2glider | |pname = 2x2glider | ||
|viewerconfig = #C [[ AUTOSTART ZOOM 32 GPS 4 TRACKLOOP 8 -1/8 1/8 ]] | |viewerconfig = #C [[ AUTOSTART ZOOM 32 GPS 4 TRACKLOOP 8 -1/8 1/8 ]] | ||
|position = right | |||
|caption = The c/8 glider | |||
|style = width:300px; | |||
|apgcode = xq8_2je4/b36s125 | |||
}} | }} | ||
===Spaceships=== | ===Spaceships=== | ||
There are a number of spaceships known to occur in 2×2.<ref>{{ | There are a number of spaceships known to occur in 2×2.<ref>{{LinkEppsteinRule|b36s125|archivedate=20150422135100|rulename=2x2 (B36/S125)}}</ref> Of these, only one is known to occur naturally from [[soup]]. It travels diagonally at c/8. | ||
===Infinite growth=== | ===Infinite growth=== | ||
The first known infinitely-growing pattern in 2×2 was discovered in June 2009 by [[Nathaniel Johnston]] while testing | The first known infinitely-growing pattern in 2×2 was discovered in June 2009 by [[Nathaniel Johnston]] while testing [[The Online Life-Like CA Soup Search]] -- a c/8 diagonal [[wickstretcher]] based on the above c/8 glider.<ref name="post531" /><ref>{{cite web|url=http://www.nathanieljohnston.com/index.php/2009/07/the-online-life-like-ca-soup-search/|title=The Online Life-Like CA Soup Search|publisher=NathanielJohnston.com|date=July 11, 2009|accessdate=July 13, 2009}}</ref> | ||
{{EmbedViewer | {{EmbedViewer | ||
|pname = 2x2linepuffer | |pname = 2x2linepuffer | ||
|viewerconfig = #C [[ AUTOSTART X -2 Y 4 ZOOM 32 GPS 4 TRACKLOOP 8 -1/8 1/8 ]] | |viewerconfig = #C [[ AUTOSTART X -2 Y 4 ZOOM 32 GPS 4 TRACKLOOP 8 -1/8 1/8 ]] | ||
|position = center | |||
|caption = The c/8 wickstretcher ({{LinkCatagolue|yl8_1_1_aae0a4678d7caeb6b463f7c082d8bd1a|rule=b36s125|style=brief|format=linear growth}}) | |||
|apgcode = yl8_1_1_aae0a4678d7caeb6b463f7c082d8bd1a/b36s125 | |||
}} | }} | ||
Multiple c/2 [[puffers]] have been discovered by [[Paul Tooke]] in {{year|2010}} including p60 forward and backward c/8 glider [[rakes]], a 2c/5 [[puffer]] was also discovered. An MMS [[breeder]] was discovered by [[Arie Paap]] on June 25, 2015.<ref name="post20740" /> On February 4, {{year|2021}}, [[FWKnightship]] constructed a [[Rule 110]] [[unit cell]] in 2×2 with these rakes, proving the rule [[Turing-complete]].<ref name="post121769" /> | |||
On February 10, {{year|2023}}, [[Luka Okanishi]] constructed a period-1024 [[gun]] for the c/8 diagonal spaceship by combining three p1024 oscillators made of {{times|2|2}} blocks. Every 1024 generations, two of the oscillators react once, and the other reacts twice for cleanup.<ref name="post157312" /> | |||
== Soup search == | |||
{{related|Catagolue|Tutorials/Contributing to Catagolue}} | |||
The {{cata|census/b36s125/C1|b36s125/C1}} census on [[Catagolue]] accumulated | |||
over 5 billion objects by April {{year|2015}},<ref group=c>[https://web.archive.org/web/20150422135936/http://catagolue.appspot.com/census /census] (archived copy as of 2015-04-22)</ref> | |||
over 415 billion objects by June {{year|2017}},<ref group=c>[https://web.archive.org/web/20170617113418/http://catagolue.appspot.com/census /census] (archived copy as of 2017-06-17)</ref> | |||
over 427 billion objects by October 2017,<ref group=c>[https://web.archive.org/web/20171029050807/http://catagolue.appspot.com/census/b36s125/C1 /census/b36s125/C1] (archived copy as of 2017-10-29)</ref> | |||
over 466 billion objects by March {{year|2020}},<ref group=c>[https://web.archive.org/web/20200312165049/https://catagolue.appspot.com/census/b36s125/C1 /census/b36s125/C1] (archived copy as of 2020-03-12)</ref> | |||
over 562 billion objects by November 2020,<ref group=c>[https://web.archive.org/web/20201119221334/https://catagolue.appspot.com/census/b36s125 /census/b36s125] (archived copy as of 2020-11-19)</ref> | |||
over 598 billion objects by December {{year|2021}},<ref group=c>[https://web.archive.org/web/20211201070951/https://catagolue.hatsya.com/census/b36s125/C1 /census/b36s125/C1] (archived copy as of 2021-12-01)</ref> | |||
over 688 billion objects by February {{year|2023}}.<ref group=c>[https://web.archive.org/web/20230221163702/https://catagolue.hatsya.com/census/b36s125/C1 /census/b36s125/C1] (archived copy as of 2023-02-21)</ref> | |||
The C1 census reached a total of 700 billion objects on April 3, 2023,<ref group=c>[https://web.archive.org/web/20230403214228/https://catagolue.hatsya.com/census/b36s125/C1 /census/b36s125/C1], [https://web.archive.org/web/20230403214439/https://catagolue.hatsya.com/haul/b36s125/C1 /haul/b36s125/C1] (archived copies as of 2023-04-03)</ref> | |||
800 billion objects on August 12, 2023.<ref group=c>[https://web.archive.org/web/20230812100356/https://catagolue.hatsya.com/census/b36s125/C1 /census/b36s125/C1], [https://web.archive.org/web/20230812100449/https://catagolue.hatsya.com/haul/b36s125/C1 /haul/b36s125/C1] (archived copies as of 2023-08-12)</ref> | |||
On September 28, {{year|2023}}, the b36s125/C1 census reached a total of one trillion objects,<ref group=c>[https://web.archive.org/web/20230928092559/https://catagolue.hatsya.com/census/b36s125/C1 /census/b36s125/C1], [https://web.archive.org/web/20230928092456/https://catagolue.hatsya.com/haul/b36s125/C1 /haul/b36s125/C1] (archived copies as of 2023-09-28)</ref> making 2×2 one of several [[Life-like rule]]s with hauls [[Catagolue#Haul verification|subject to statistical verification and peer review]] before being committed. | |||
== See also == | == See also == | ||
*[[List of Life-like cellular automata]] | * [[List of Life-like cellular automata]] | ||
* | * {{rl|HighFlock}} | ||
* {{rl|Flock}} | |||
==References== | ==References== | ||
<references /> | <references> | ||
<ref name="post531">{{LinkForumThread | |||
|format = ref | |||
|title = First infinite growth in 2x2 (B36/S125)? | |||
|p = 531 | |||
|author = Nathaniel Johnston | |||
|date = June 29, 2009 | |||
}}</ref> | |||
<ref name="post20740">{{LinkForumThread | |||
|format = ref | |||
|title = Re: 2x2 | |||
|p = 20740 | |||
|author = Arie Paap | |||
|date = June 25, 2015 | |||
}}</ref> | |||
<ref name="post121769">{{LinkForumThread | |||
|format = ref | |||
|title = Re: List of the Turing-complete totalistic life-like CA | |||
|p = 121769 | |||
|author = FWKnightship | |||
|date = February 4, 2021 | |||
}}</ref> | |||
<ref name="post157312">{{LinkForumThread | |||
|format = ref | |||
|title = Re: 2x2 | |||
|p = 157312 | |||
|author = Luka Okanishi | |||
|date = February 10, 2023 | |||
}}</ref> | |||
</references> | |||
Catagolue: | |||
<references group=c> | |||
</references> | |||
==Further reading== | ==Further reading== | ||
* [[Nathaniel Johnston]], ''[https://arxiv.org/abs/1203.1644 The B36/S125 "2x2" Life-Like Cellular Automaton]'', in: | * [[Nathaniel Johnston]], ''[https://arxiv.org/abs/1203.1644 The B36/S125 "2x2" Life-Like Cellular Automaton]'', in: Andrew Adamatzky (ed.), ''[[Game of Life Cellular Automata]]'', Springer 2010, pp. 99-114 | ||
==External links== | ==External links== | ||
{{LinkForumThread|f=11|t=39|title=2x2}} | * {{LinkForumThread|f=11|t=39|title=2x2}} | ||
{{LinkCatagolueRule|b36s125}} | * {{LinkCatagolueRule|b36s125}} | ||
{{LinkEppsteinRule|b36s125}} | * {{LinkEppsteinRule|b36s125|archivedate=20150422135100}} | ||
* {{LinkMirek|rullex_life.html#2x2|title=MCell built-in Life rules: 2x2}} | |||
[[Category:Pages using linear growth apgcodes]] | |||
[[Category:Life-like cellular automata]] | |||
[[Category:2×2 block rules]] |
Latest revision as of 15:29, 30 December 2023
2×2 | |
View static image | |
View animated image | |
Rulestring | 125/36 B36/S125 |
---|---|
Rule integer | 19528 |
Character | Chaotic |
Black/white reversal | B012458/S0134678 |
2×2 is a Life-like cellular automaton in which cells survive from one generation to the next if they have 1, 2 or 5 neighbours, and are born if they have 3 or 6 neighbours. It thus has rulestring "B36/S125". Patterns under the rule have a chaotic evolution similar to those under the standard Life, but the chaos tends to die out much more quickly.
Its name comes from the fact that patterns made up of 2 × 2 blocks continue to evolve as patterns made up of 2 × 2 blocks.
Block evolution
The 2×2 rule can emulate a simpler cellular automaton that acts on each 2 × 2 block. The emulated automaton is a block cellular automaton that makes use of the Margolus neighbourhood and evolves according to the following six rules:
Note that, as this emulates a Margolus neighbourhood, the resulting block appears at the center of the original four blocks. Thus, patterns that are originally made up of 2 × 2 blocks will forever be made up of 2 × 2 blocks, but the block partition will be offset by one cell in the odd generations from the even generations. By examining the image above, one can see that an isotropic cellular automaton will emulate a Margolus block cellular automaton if and only if the following four equations are satisfied: B4w = S4q, B5a = S6a = S7c, B3i = S5i, B1c = B2a = S3a, where the first equation for example means that the birth condition for cells with four neighbours must equal the survival condition for cells with four neighbours. There are 212 = 4096 such rules, which emulate 26 = 64 different block cellular automata. Any arrangement of cells that fits within a 2 × 2 bounding box can simulate these using isotropic non-totalistic rules.
This rule can be seen to satisfy the above equations because 4 is neither a birth condition nor a survival condition, 5 is not a birth condition and 6 and 7 are not survival conditions, 3 is a birth condition and 5 is a survival condition, and 3 is not a survival condition and 1 and 2 are not birth conditions.
The non-totalistic cellular automaton B3i4int5ey6k7e/S1e2k3ey4irt5i can be used to simulate this rule. 1 × 1 cells simulate the clusters of 2 × 2 blocks, and only every second generation plays, since odd generations have the offset. Since this rule is self-similar when scaling up (patterns made of 4 × 4 blocks will remain made of 4 × 4 blocks every second generation, 8 × 8 every fourth, etc.) this rule itself can simulate the mentioned 2 × 2 oscillators.
Notable patterns
A large variety of still lifes and oscillators appear spontaneously from randomly generated starting states. There is also a somewhat rare naturally-occurring spaceship, which travels at c/8 diagonally.
Still lives
Still lives are generally smaller in 2×2 than in Life, with the smallest occurring having a population of just 2 cells. These still life patterns still tend to be similar to Life patterns in terms of structure, for example often having islands that stabilise each other. Many still lives from Life are also still lives in 2×2, for example, the beehive, tub, loaf, pond and mango.
Some sample still lifes (click above to open LifeViewer) RLE: here Plaintext: here |
Enumerating still lives
The following table catalogs all still lives in the 2×2 rule with 10 or fewer cells.[1]
Size | Count | Image | Links |
---|---|---|---|
1 | 0 | ||
2 | 2 | Download RLE: click here | |
3 | 1 | Download RLE: click here | |
4 | 3 | Download RLE: click here | |
5 | 4 | Download RLE: click here | |
6 | 9 | Download RLE: click here | |
7 | 10 | Download RLE: click here | |
8 | 27 | Download RLE: click here | |
9 | 48 | Download RLE: click here | |
10 | 126 | Download RLE: click here |
Common still lives
The following table lists the twenty most common strict still lifes that arise after several generations of a random starting pattern.[2] The "approx. rel. freq." column gives an estimate of the proportion of all randomly-occurring still lives that will be of the given type.
|
|
Oscillators
A large variety of oscillators of various periods occur naturally in 2×2.
Period two oscillators
Many of the period 2 oscillators in 2×2 have a single-cell 'on-off' rotor, with small variations in the stator of the oscillator. These occur fairly frequently naturally.
Some period 2 oscillators (click above to open LifeViewer) RLE: here Plaintext: here |
Higher-period oscillators
One of the most interesting aspects of the 2×2 rule is the large number of naturally-occurring higher-period oscillators. Oscillators with periods 3, 4, 5, 6, 10, 14, 22 and 26 are all relatively frequent, and oscillators are also known for periods 7, 8, 11, 12, 17, 20, 24, 28, 30, 60, 62 and 126.
A stamp collection of oscillators with different periods from 2 through 126 (click above to open LifeViewer) RLE: here Plaintext: here |
One simple infinite family of oscillators is given by the 2 × (4n) boxes of alive cells.[3] Such oscillators can be analyzed by noting that each phase of their oscillation can be represented as an exclusive or (XOR) of rectangles of different sizes that emulate the Rule 90 cellular automaton.[4] The period of these oscillators for n = 1, 2, 3, ... is given by the sequence 2, 6, 14, 14, 62, 126, 30, 30, 1022, ... (Sloane's A160657).
Naturally occurring oscillators
The following table lists the twenty most common oscillators that arise after several generations of a random starting pattern.[2] Of particular interest are some quite high-period oscillators that appear abnormally frequently (in particular, the period 26 stairstep hexomino is the third most common oscillator). The "approx. rel. freq." column gives an estimate of the proportion of all randomly-occurring oscillators that will be of the given type.
|
|
Spaceships
There are a number of spaceships known to occur in 2×2.[5] Of these, only one is known to occur naturally from soup. It travels diagonally at c/8.
Infinite growth
The first known infinitely-growing pattern in 2×2 was discovered in June 2009 by Nathaniel Johnston while testing The Online Life-Like CA Soup Search -- a c/8 diagonal wickstretcher based on the above c/8 glider.[6][7]
The c/8 wickstretcher (Catagolue: here) (click above to open LifeViewer) RLE: here Plaintext: here Catagolue: here |
Multiple c/2 puffers have been discovered by Paul Tooke in 2010 including p60 forward and backward c/8 glider rakes, a 2c/5 puffer was also discovered. An MMS breeder was discovered by Arie Paap on June 25, 2015.[8] On February 4, 2021, FWKnightship constructed a Rule 110 unit cell in 2×2 with these rakes, proving the rule Turing-complete.[9]
On February 10, 2023, Luka Okanishi constructed a period-1024 gun for the c/8 diagonal spaceship by combining three p1024 oscillators made of 2 × 2 blocks. Every 1024 generations, two of the oscillators react once, and the other reacts twice for cleanup.[10]
Soup search
- See also: Catagolue, Tutorials/Contributing to Catagolue
The b36s125/C1 census on Catagolue accumulated over 5 billion objects by April 2015,[c 1] over 415 billion objects by June 2017,[c 2] over 427 billion objects by October 2017,[c 3] over 466 billion objects by March 2020,[c 4] over 562 billion objects by November 2020,[c 5] over 598 billion objects by December 2021,[c 6] over 688 billion objects by February 2023.[c 7] The C1 census reached a total of 700 billion objects on April 3, 2023,[c 8] 800 billion objects on August 12, 2023.[c 9]
On September 28, 2023, the b36s125/C1 census reached a total of one trillion objects,[c 10] making 2×2 one of several Life-like rules with hauls subject to statistical verification and peer review before being committed.
See also
References
- ↑ Computed using the EnumStillLifes.c script located here.
- ↑ 2.0 2.1 Full results are located here.
- ↑ Nathaniel Johnston (May 22, 2009). "Rectangular Oscillators in the 2×2 (B36/S125) Cellular Automaton". Retrieved on May 24, 2009.
- ↑ "Life 2x2: long oscillator". comp.theory.cell-automata (November 2, 2001). Retrieved on May 24, 2009.
- ↑ 2x2 (B36/S125) at David Eppstein's Glider Database
- ↑ Nathaniel Johnston (June 29, 2009). First infinite growth in 2x2 (B36/S125)? (discussion thread) at the ConwayLife.com forums
- ↑ "The Online Life-Like CA Soup Search". NathanielJohnston.com (July 11, 2009). Retrieved on July 13, 2009.
- ↑ Arie Paap (June 25, 2015). Re: 2x2 (discussion thread) at the ConwayLife.com forums
- ↑ FWKnightship (February 4, 2021). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums
- ↑ Luka Okanishi (February 10, 2023). Re: 2x2 (discussion thread) at the ConwayLife.com forums
Catagolue:
- ↑ /census (archived copy as of 2015-04-22)
- ↑ /census (archived copy as of 2017-06-17)
- ↑ /census/b36s125/C1 (archived copy as of 2017-10-29)
- ↑ /census/b36s125/C1 (archived copy as of 2020-03-12)
- ↑ /census/b36s125 (archived copy as of 2020-11-19)
- ↑ /census/b36s125/C1 (archived copy as of 2021-12-01)
- ↑ /census/b36s125/C1 (archived copy as of 2023-02-21)
- ↑ /census/b36s125/C1, /haul/b36s125/C1 (archived copies as of 2023-04-03)
- ↑ /census/b36s125/C1, /haul/b36s125/C1 (archived copies as of 2023-08-12)
- ↑ /census/b36s125/C1, /haul/b36s125/C1 (archived copies as of 2023-09-28)
Further reading
- Nathaniel Johnston, The B36/S125 "2x2" Life-Like Cellular Automaton, in: Andrew Adamatzky (ed.), Game of Life Cellular Automata, Springer 2010, pp. 99-114
External links
- 2x2 (discussion thread) at the ConwayLife.com forums
- 2×2 at Adam P. Goucher's Catagolue
- 2×2 at David Eppstein's Glider Database
- MCell built-in Life rules: 2x2 at Mirek Wójtowicz's Cellebration page