OCA:2×2

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2×2
2×2 rule
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 2x2 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 Thus, patterns that are originally made up of 2×2 blocks will forever be made up of 2×2 blocks, but the block partition will be offset by one cell in the odd generations from the even generations.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 2x2 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. 1x1 cells simulate the clusters of 2x2 blocks, and only every second generation plays, since odd generations have the offset.

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 2x2 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.

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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.
(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 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)
(click above to open LifeViewer)
RLE: here Plaintext: here

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)
(click above to open LifeViewer)
RLE: here Plaintext: here

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