Difference between revisions of "OCA:2×2"

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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:
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:


[[Image:2x2block_evolve.png|framed|center|The 2x2 block evolution rule]]
[[Image:2x2block_evolve.png|framed|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 Thus, patterns that are originally made up of 2&times;2 blocks will forever be made up of 2&times;2 blocks, but the block partition will be offset by one [[cell]] in the odd [[generation]]s from the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 a Life-like cellular automaton will emulate a Margolus block cellular automaton if and only if the following four equations are satisfied: B4 = S4, B5 = S6 = S7, B3 = S5, B1 = B2 = S3, 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 2<sup>12</sup> = 4096 such rules, which emulate 2<sup>6</sup> = 64 different block cellular automata. Any arrangement of cells that fits within a 2x2 bounding box can simulate these using isotropic non-totalistic 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 Thus, patterns that are originally made up of 2&times;2 blocks will forever be made up of 2&times;2 blocks, but the block partition will be offset by one [[cell]] in the odd [[generation]]s from the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 a Life-like cellular automaton will emulate a Margolus block cellular automaton if and only if the following four equations are satisfied: B4 = S4, B5 = S6 = S7, B3 = S5, B1 = B2 = S3, 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 2<sup>12</sup> = 4096 such rules, which emulate 2<sup>6</sup> = 64 different block cellular automata. Any arrangement of cells that fits within a 2&times;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 Life-like cellular automaton]] B3i4int5ey6k7e/S1e2k3ey4irt5i can be used to simulate this rule. 1x1 cells simulate the clusters of 2x2 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 4x4 blocks will remain made of 4x4 blocks every second generation, 8x8 every fourth, etc.) this rule itself can simulate the mentioned 2x2 oscillators.
The [[non-totalistic Life-like cellular automaton]] B3i4int5ey6k7e/S1e2k3ey4irt5i can be used to simulate this rule. 1&times;1 cells simulate the clusters of 2&times;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&times;4 blocks will remain made of 4&times;4 blocks every second generation, 8&times;8 every fourth, etc.) this rule itself can simulate the mentioned 2&times;2 oscillators.


==Notable patterns==
==Notable patterns==
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====Enumerating still lifes====
====Enumerating still lifes====
The following table catalogs all still lifes in the 2x2 rule with 10 or fewer [[cell]]s.<ref>Computed using the ''EnumStillLifes.c'' script located [http://www.conwaylife.com/forums/viewtopic.php?f=9&t=44 here].</ref>
The following table catalogs all still lifes in the 2&times;2 rule with 10 or fewer [[cell]]s.<ref>Computed using the ''EnumStillLifes.c'' script located [http://www.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;"

Revision as of 21:39, 3 February 2020

2×2
x=0, y = 0, rule = B36/S125 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 RANDSEED 1729 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]]
LifeViewer-generated pseudorandom soup
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 rules, 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:

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 2×2 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 a Life-like cellular automaton will emulate a Margolus block cellular automaton if and only if the following four equations are satisfied: B4 = S4, B5 = S6 = S7, B3 = S5, B1 = B2 = S3, 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 Life-like 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 lifes

Still lifes 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 lifes from Life are also still lifes in 2×2, For example, the beehive, tub, loaf, pond and mango.

x = 38, y = 12, rule = B36/S125 19bo10b2o6b$12bo5bo5bo9b2o2b$6bo4bo5bo4b3o3b4o6b$bo3bo4bo5bo6b3o12b$o 3bo4bo5bo7bo4b2o6bob$35bobo$35bo2b$bo4bo4b2o4b2o5bo8bobo2b$obo2bobo2bo 2bo2bo2bo3bobo3b2o2bobobo$bo3bobo3bobo2bo2bo2bo2bo2bo7bob$6bo5bo4b2o3b obo4b2o7b$23bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
Some sample still lifes.
(click above to open LifeViewer)
RLE: here Plaintext: here

Enumerating still lifes

The following table catalogs all still lifes in the 2×2 rule with 10 or fewer cells.[1]

Size Count Image Links
1 0
2 2 2x22cellstilllifes.png Download RLE: click here
3 1 2x23cellstilllifes.png Download RLE: click here
4 3 2x24cellstilllifes.png Download RLE: click here
5 4 2x25cellstilllifes.png Download RLE: click here
6 9 2x26cellstilllifes.png Download RLE: click here
7 10 2x27cellstilllifes.png Download RLE: click here
8 27 2x28cellstilllifes.png Download RLE: click here
9 48 2x29cellstilllifes.png Download RLE: click here
10 126 2x210cellstilllifes.png Download RLE: click here

Common still lifes

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 lifes that will be of the given type.

Rank Pattern # of cells Approx. rel. freq. (out of 1.00)
1 2x2 stilllife rank1.png (domino) 2 0.582
2 2x2 stilllife rank2.png 2 0.251
3 2x2 stilllife rank3.png 5 0.052
4 2x2 stilllife rank4.png 3 0.0498
5 2x2 stilllife rank5.png 6 0.0252
6 2x2 stilllife rank6.png 4 0.019
7 2x2 stilllife rank7.png 5 0.00725
8 2x2 stilllife rank8.png (beehive) 6 0.00384
9 2x2 stilllife rank9.png (tub) 4 0.00322
10 2x2 stilllife rank10.png 5 0.00195
Rank Pattern # of cells Approx. rel. freq. (out of 1.00)
11 2x2 stilllife rank11.png 4 0.00124
12 2x2 stilllife rank12.png (loaf) 7 5.8×10-4
13 2x2 stilllife rank13.png 6 5.63×10-4
14 2x2 stilllife rank14.png 6 4.04×10-4
15 2x2 stilllife rank15.png 7 2.56×10-4
16 2x2 stilllife rank16.png (aircraft carrier) 6 2.23×10-4
17 2x2 stilllife rank17.png (pond) 8 1.94×10-4
18 2x2 stilllife rank18.png (mango) 8 1.28×10-4
19 2x2 stilllife rank19.png 5 9.6×10-5
20 2x2 stilllife rank20.png 6 7.68×10-5

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.

x = 51, y = 16, rule = B36/S125 49b2o$34b2o11b2obo$27b2o4bobo4bo5bobo2b$2b2o3b2o4b2o4b3o4bo13bo5bo4b$ 3bo3b2o6bo2bobobo2bobo3bobo4bobobo4bobob$b2o4b2o3b2o4bobobo2bo5b2o5b2o 2bo5b2ob$7b2o42b3$37b2o6b2o4b$2b2o5b2o7bo7bo2bo6bo2bo5bo5b$3bo5bob2o5b ob2o5b3o4b3obo4bob4o2b$o3b2o5b2o7bobo3b3o4bo9b4obo2b$2o3bo5b2obo3bob2o 4bo2bo4b2o10bo4b$2bo10b2o3bo26b2o4b$2b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 AUTOSTART GPS 2 ]]
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 8, 11, 12, 17, 24 and 60.

x = 277, y = 175, rule = B36/S125 3b2o272b2$6bo86bo4bo95b2o81b$6bo86bo4bo63bo2b2o37b3o70b$68bo6bo85bobo 4bo23bo4bo6bo5b2o65b$3b2o34b2o28bo4bo20b2o7b2o5bobo20b5o13bo7bo2bo2b2o 24bo4bo7bob3obo65b$25b2o14bo5b2ob2o2b2o8b2o5bo33b2o4bo6b2o6b2o6b5o7bo 3b2obo5bo2bobo2bobo35bo5bo65b$bo5b4o3b2o3b2o4bobo5b3o4bobo4bo3bo2bobo 8bo4b3o25bo8bo3bo4bo3bo4b2obo3b2o5b2o4bo7bo6b2obo5bo24b2o9bo3bo67b$bo 5b4o4bo5bo6bo3bobobo4bo6bobo6b2o3b3o6b3o24bo7b2o3b2o3b2o2bo3b2ob2o5b5o 6bob2o3bo9bo3bo36bo5bo67b$13b2o3b2o7b2o3bobobo5b2o5bo4bobo5b3o7bo61b5o 7bo12bo5bob2o23bo4bo5bob3obo67b$3b2o43b2o4b2o5bo7bo4bo23bo60bobo2bobo 2bo22bo4bo5b2o5bo66b$61b2o5bo6bo22bo62b2o2bo2bo39b3o66b$160bo4bobo26b 2o81b$106bo2b2o14b4o14b2o6bo4bo5b2o2bo110b$11b4o5b2o83bo2bobo3bo3bo6b 4o14bobo6bo2bo121b$11b4o5b2o10b2o38b2o34b2o4b2ob2o6b4o7bo2bo5b2o6bo 123b$3b2o15b2o11bo2b2ob2o2b2o9b2o3b2o2bo10bo3b2o2b2o2bo19b2o7bobo7b4o 18bo4b3o6bo4bo110b$3bob2o13b2o8bobo3bo3bo2bo9bo2bo2bobobo5bob3o4bobobo bobo18bobo2bo3b2ob2o6b4o6bob2obo6bo4bob2o5b2o2b2o110b$5bobo5b4o5b4o2bo bo6b3o4b3o7bo6bo6bo2bo8bo3bo20b2o2bo4bo3bo6b4o6bo4bo4b2o5b3o7bo2bo111b $7bo5b4o5b4o5bo15bo4bobobo2bobobo5b2o7bobobobobo38b4o7b4o3bobo7bo8bo2b o111b$4bobo23b2o4b2ob2o5b2o4bo2b2o2bo2b2o14b2o2b2o2bo38b4o14b2o7bo2bo 5b2o2b2o31b2o39bo4bo32b$4b2o145bo4bo4bo4bo26bo28b2o15bo4bo32b$193bo2bo 4bo19b4o52b$196bo4bo7b2ob2o6bob2obo13bo4bo7b2o23b$67b2o3b2o119bo17bo8b o4bo13bo4bo10b2o20b$34b4o29b2o3b2o33b2ob2o30bo7bo42bo2bo4bo5bo7bo6b2o 31b2obo18b$7b2obo5b2o16b4o7b2o20b2o3b2o32bobobobo30bo5bo46bo4bo9bo10b 2o15bo4bo10bo3bo17b$8bobo6bo6bo2bo4b2o4b2o5bo7b2o5b2o5b2o3b2o32b2o3b2o 11bo5bo15bo46bo15b2ob2o6bo4bo13bo4bo11bob2o17b$8b2o4b4obo5b3o4b2o4b2o 3b2o3bo5bo2bo2bo8b3o35b2ob2o10bobo5bobo12b3o14bo30bo2bo4bo18bob2obo32b 2o17b$4bo2b2o5bob4o4b3o5b2o4b2o3bo3b2o4bo2bobo2bo5b2o3b2o29b2o3b3o3b2o 7bo7bo6bo4b3ob3o4bo4b2o3bo2bo28bo4bo19b4o14bo4bo16b2o14b$4b5o7bo7bo2bo 4b2o4b2o6bo7bo2bo2bo6b2o3b2o28bob2o3bo3b2obo4b2o3b3o3b2o5bo2bo2b3o2bo 2bo5bo4bobo26bo28b2o15bo4bo32b$6bo9b2o16b4o7b2o6b2o5b2o5b2o3b2o28b2ob 2o5b2ob2o8b2ob2o12bo7bo10b2o2bo27bo4b2o77b$4b2o28b4o29b2o3b2o32b2o3b2o 11b7o10b3o5b3o6bo2bo2bob2o109b$102b2ob2o5b2ob2o7bobobobo9b2obo5bob2o 124b$102bob2o3bo3b2obo7b7o10b3o5b3o7b2obo2bo2bo108b$103b2o3b3o3b2o9b2o b2o12bo7bo10bo2b2o111b$8b2ob2o5bo4bo3bo4bo5bo26b2ob2o37b2ob2o9b2o3b3o 3b2o5bo2bo2b3o2bo2bo6bobo4bo109b$4b2o6bo6bo4bobo4bo3bobobo8b2o4b2o8bo 41b2o3b2o10bo7bo6bo4b3ob3o4bo4bo2bo3b2o109b$4bobob4o9bo10b2o2b2o5bobob o6bobo6bobob3obo34bobobobo9bobo5bobo12b3o16bo45bo2bo14b5o44b$6bobo10bo bob3o7bo9bob5o6bobo4bo2bo2bobo35b2ob2o12bo5bo15bo46bo4b2o9bob2obo13b5o 44b$4bobo3b2obo4bo2bo10bo5bo5b2o2bo5bo2bo7bobobo73bo5bo43bo16bo2bo12b 2o5b2o42b$4bob2o3bobo10bo2bo9bo7bo2b2o3bo4b2o3bobo2bo2bo70bo7bo50bo7bo b2obo11b2o5b2o42b$9bobo8b3obobo6b2o2b2o5b5obo9bo2bob3obobo121bo7bo4bob obo2bobobo8b2o5b2o33bo4b2o2b$6b4obobo10bo6bobobo3bo6bobobo6bob2o9bo 122bo11bobobo4bobobo7b2o5b2o14bo2b2o4bo5bo4bo2bo3bo$5bo6b2o5bobo4bo5bo 5bo5b2o10bo8b2ob2o89b2o3bo4bo28b2o6bobo6bobo8b2o5b2o10bo3bo4bo2bo2bo4b 2o2b4ob2ob$5b2ob2o8bo3bo4bo133bo2b4obo23bo12bobo6bobo8b2o5b2o11bobob3o 2bobo4bo2bo3bob2o4b$105b2obo3bob2o2bobobo2bobobo2bobobo5bo2bo4bo2bo6bo 3b3obo24bo2bo8bobobo4bobobo7b2o5b2o7b2o6bobo2bobo2bo4b2o5bo4b$105bob2o 3b2obo2b2ob2o2b2ob2o2b2ob2o5bo3bo2bo3bo6b2o2bo31bo9bobobo2bobobo8b2o5b 2o11bo3bo3b3obo5bo5b2o5b$21bo24bo60bobobobo32b4o10b2o31bo15bob2obo11b 2o5b2o10bo4b2o2bo22b$4b2o2bo2b2o7bobo9bo5bo6bo15bo5b2o38bo2bo2bo6bo6bo 6bo9bo6bo7b4o3b4o23bo4b2o10bo2bo12b2o5b2o42b$4bo4bo2bo9bo8bobo5bo3b3o 2bo7b2o2bo9bo37b2ob2o6bo8bo6bo9b6o16b2o40bob2obo13b5o44b$6bob3o8bob3o 9b3ob2o7bobo6bo2bo8b2o51bo7bo5bo29bo2b2o41bo2bo14b5o44b$5b2o10b3o5bo 11bo3bo2b3ob2o4bobob2o8bob2o2bobobo29b2ob2o14bo6bo25bob3o3bo108b$4bobo 3bobo3bo2bo3bo2bo6bo3bo4b2ob3o2bo3bo5bo5bobo7bobo28bo2bo2bo4b2ob2o10bo 11b6o8bob4o2bo109b$10b2o5bo5b3o7bo9bobo9b2obobo5bobobo2b2obo30bobobobo 4bobobo2b2ob2o4bo9bo6bo7bo4bo3b2o107b$6b3obo8b3obo8b2ob3o5bo2b3o7bo2bo 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6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 HEIGHT 720 WIDTH 960 ]]
A stamp collection of oscillators with different periods from 2 through 60.

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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 OEISicon light 11px.pngA160657).

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.

Rank Pattern Period Minimum # of cells Approx. rel. freq. (out of 1.00)
1 2x2 oscillator rank1.gif 2 5 0.494
2 2x2 oscillator rank2.gif 2 8 0.204
3 2x2 oscillator rank3.gif 26 6 0.0698
4 2x2 oscillator rank4.gif 2 5 0.0514
5 2x2 oscillator rank5.gif 4 6 0.0332
6 2x2 oscillator rank6.gif 14 7 0.0324
7 2x2 oscillator rank7.gif 4 6 0.0285
8 2x2 oscillator rank8.gif 2 6 0.0217
9 2x2 oscillator rank9.gif 4 6 0.0169
10 2x2 oscillator rank10.gif 4 7 0.0152
Rank Pattern Period Minimum # of cells Approx. rel. freq. (out of 1.00)
11 2x2 oscillator rank11.gif 2 8 0.00848
12 2x2 oscillator rank12.gif 2 6 0.007
13 2x2 oscillator rank13.gif 10 12 0.00457
14 2x2 oscillator rank14.gif 2 7 0.00196
15 2x2 oscillator rank15.gif 2 7 0.00175
16 2x2 oscillator rank16.gif 2 6 0.00175
17 2x2 oscillator rank17.gif 14 6 0.00156
18 2x2 oscillator rank18.gif 2 8 0.00106
19 2x2 oscillator rank19.gif 6 16 0.00106
20 2x2 oscillator rank20.gif 22 8 0.00043
x = 5, y = 4, rule = B36/S125 3bo$obo2$2b3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 GPS 4 TRACKLOOP 8 -1/8 1/8 ]]
The c/8 glider (Catagoluehere)

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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 at c/8 diagonally.

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] 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. No guns have yet been discovered in 2×2. An MMS breeder was discovered by Arie Paap on June 25, 2015.

x = 11, y = 15, rule = B36/S125 10bo$9bo$8bo$7bo$6bo$5bo$4bo$3bo$2bo2$o$o2bo$obo2$2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART X -2 Y 4 ZOOM 32 GPS 4 TRACKLOOP 8 -1/8 1/8 ]]
The c/8 wickstretcher (Catagoluehere)

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See also

References

  1. Computed using the EnumStillLifes.c script located here.
  2. 2.0 2.1 Full results are located here.
  3. Nathaniel Johnston (May 22, 2009). "Rectangular Oscillators in the 2×2 (B36/S125) Cellular Automaton". Retrieved on May 24, 2009.
  4. "Life 2x2: long oscillator". comp.theory.cell-automata (November 2, 2001). Retrieved on May 24, 2009.
  5. "2x2 (B36/S125)". David Eppstein. Retrieved on March 18, 2009.
  6. "First infinite growth in 2x2 (B36/S125)?". ConwayLife.com forums. Retrieved on July 13, 2009.
  7. "The Online Life-Like CA Soup Search". NathanielJohnston.com (July 11, 2009). Retrieved on July 13, 2009.

Further reading

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