OCA:2×2

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
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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:

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 Thus, patterns that are originally made up of 2 × 2 blocks will forever be made up of 2 × 2 blocks, but the block partition will be offset by one cell in the odd generations from the even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 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.

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 lives

The following table catalogs all still lives in the 2×2 rule with 10 or fewer cells.

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. The "approx. rel. freq." column gives an estimate of the proportion of all randomly-occurring still lives that will be of the given type.

Rank Pattern # of cells Approx. rel. freq. (out of 1.00) 1 (domino) 2 0.582 2 (duoplet) 2 0.251 3 5 0.052 4 3 0.0498 5 6 0.0252 6 4 0.019 7 5 0.00725 8 (beehive) 6 0.00384 9 (tub) 4 0.00322 10 5 0.00195

Rank Pattern # of cells Approx. rel. freq. (out of 1.00) 11 4 0.00124 12 (loaf) 7 5.8 × 10-4 13 6 5.63 × 10-4 14 6 4.04 × 10-4 15 7 2.56 × 10-4 16 (aircraft carrier) 6 2.23 × 10-4 17 (pond) 8 1.94 × 10-4 18 (mango) 8 1.28 × 10-4 19 5 9.6 × 10-5 20 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 7, 8, 11, 12, 17, 20, 24, 28, 30, 60, 62 and 126.

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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. 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. 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. 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 2 5 0.494 2 2 8 0.204 3 26 6 0.0698 4 2 5 0.0514 5 4 6 0.0332 6 14 7 0.0324 7 4 6 0.0285 8 2 6 0.0217 9 4 6 0.0169 10 4 7 0.0152

Rank Pattern Period Minimum # of cells Approx. rel. freq. (out of 1.00) 11 2 8 0.00848 12 2 6 0.007 13 10 12 0.00457 14 2 7 0.00196 15 2 7 0.00175 16 2 6 0.00175 17 14 6 0.00156 18 2 8 0.00106 19 6 16 0.00106 20 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
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagolue: here

Spaceships

There are a number of spaceships known to occur in 2×2. 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.

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 (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. On February 4, 2021, FWKnightship constructed a Rule 110 unit cell in 2×2 with these rakes, proving the rule Turing-complete.

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.

Soup search

See also: Catagolue, Tutorials/Contributing to Catagolue

The b36s125/C1 census on Catagolue accumulated over 5 billion objects by April 2015, over 415 billion objects by June 2017, over 427 billion objects by October 2017, over 466 billion objects by March 2020, over 562 billion objects by November 2020, over 598 billion objects by December 2021, over 688 billion objects by February 2023. The C1 census reached a total of 700 billion objects on April 3, 2023, 800 billion objects on August 12, 2023.

On September 28, 2023, the b36s125/C1 census reached a total of one trillion objects, making 2×2 one of several Life-like rules with hauls subject to statistical verification and peer review before being committed.

See also

• List of Life-like cellular automata
• HighFlock
• Flock

References

Catagolue:

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