# OCA:2×2

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 Thus, patterns that are originally made up of 2×2 blocks will forever be made up of 2×2 blocks, but the block partition will be offset by one cell in the odd generations from the even generations.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

 Please enable Javascript to view this LifeViewer. 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]

1 0

### 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.

Rank Pattern # of cells Approx. rel. freq. (out of 1.00)
1 (domino) 2 0.582
2 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.

 Please enable Javascript to view this LifeViewer. 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.

 Please enable Javascript to view this LifeViewer. 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 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.

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
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 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] 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.[8]

On February 4, 2021, FWKnightship constructed a Rule 110 unit cell in 2×2 with the aforementioned rakes, proving the rule Turing-complete.[9]

 Please enable Javascript to view this LifeViewer. The c/8 wickstretcher (Catagolue: here) (click above to open LifeViewer)RLE: here Plaintext: here

## References

1. Computed using the EnumStillLifes.c script located here.
2. 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. Nathaniel Johnston (June 29, 2009). First infinite growth in 2x2 (B36/S125)? (discussion thread) at the ConwayLife.com forums
7. "The Online Life-Like CA Soup Search". NathanielJohnston.com (July 11, 2009). Retrieved on July 13, 2009.
8. Arie Paap (June 25, 2015). Re: 2x2 (discussion thread) at the ConwayLife.com forums
9. FWKnightship (February 4, 2021). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums