## Smallest Linear Replicators Supporting Specific Speeds and Habits

For discussion of other cellular automata.
muzik
Posts: 3774
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Smallest Linear Replicators Supporting Specific Speeds and Habits

Over the history of OCA research there's been a multitude of interesting linear and quadratic replicators discovered. This project is to serve as a means to consolidate all of the discovered replicators.

Currently this project has the same constraints as the 5s project - only outer-totalistic or isotropic non-totalistic range-1 2-state rules, with alternating and B0 also excluded, however this may be expanded in the future.

In addition to a displacement vector and period, replication habit is also something that must be taken into account for linear replicators - they preferably should act as a 0E0P-like unit cell of the rule they replicate, so that any amount of correctly spaced copies still function like an isolated one would. The notation is as follows:

(x1,y1,x2,y2)c/p kK Ff U
where
x1 = horizontal displacement of a replicator section
y1 = vertical displacement of a replicator section
x2 = overall horizontal displacement of the replicator
y2 = overall vertical displacement of the replicator
p = period
K = number of states in 1D parity rule
F = number of copies present at generation p
U = present if the replicator functions as a unit cell of a 1D parity rule - removing copies of the replicator will still result in it replicating in the same way in future
(this notation may need some refining though)

Here are some examples:

(1,0,0,0)c/1 k2 2f U

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``````x = 1, y = 2, rule = B2a/S
o\$o!``````

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``````x = 15, y = 2, rule = B2a/S
obobobobobo3bo\$obobobobobo3bo!``````
(4,0,0,0)c/4 k2 3f U

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``````x = 1, y = 5, rule = B2a/S1e3eiy5i
o\$o2\$o\$o!``````

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``````x = 57, y = 5, rule = B2a/S1e3eiy5i
o3bo7bo3bo7bo3bo3bo7bo3bo11bo\$o3bo7bo3bo7bo3bo3bo7bo3bo11bo2\$o3bo7bo3b
o7bo3bo3bo7bo3bo11bo\$o3bo7bo3bo7bo3bo3bo7bo3bo11bo!``````
(4,0,0,0)c/11 k2 2f U

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``````x = 5, y = 3, rule = B36/S2-i34q
5o\$5o\$5o!``````

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``````x = 61, y = 3, rule = B36/S2-i34q
5o3b5o3b5o3b5o3b5o3b5o11b5o\$5o3b5o3b5o3b5o3b5o3b5o11b5o\$5o3b5o3b5o3b5o
3b5o3b5o11b5o!``````
(2,2,0,0)c/12 k2 2f U

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``````x = 5, y = 5, rule = B36/S23
2b3o\$bo2bo\$o3bo\$o2bo\$3o!``````

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``````x = 33, y = 33, rule = B36/S23
2b3o\$bo2bo\$o3bo\$o2bo\$3o3b3o\$5bo2bo\$4bo3bo\$4bo2bo\$4b3o3b3o\$9bo2bo\$8bo3b
o\$8bo2bo\$8b3o3b3o\$13bo2bo\$12bo3bo\$12bo2bo\$12b3o3b3o\$17bo2bo\$16bo3bo\$
16bo2bo\$16b3o3b3o\$21bo2bo\$20bo3bo\$20bo2bo\$20b3o4\$30b3o\$29bo2bo\$28bo3bo
\$28bo2bo\$28b3o!``````
(5,0,0,0)c/10 k3 2f

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``````x = 3, y = 4, rule = B3-e4ny56i7e/S2-ck3r4-nt5jq6
bo\$3o\$3o\$bo!``````
(7,0,0,0)c/14 k4 2f

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``````x = 4, y = 3, rule = B2ei3-akny4acnqrw5aqy6aen/S1c2-ik3ack4cjknrw5acjk6c7
b2o\$o2bo\$b2o!``````
(7,0,0,0)c/14 k5 2f

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``````x = 4, y = 3, rule = B2e3-cnqy4rwyz5y6ace7c/S1c2ac3aiq4eiknrwy5cn6ekn7c8
b2o\$o2bo\$b2o!``````
(2,0,2,2)c/2 k2 2f U

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`````` x = 2, y = 2, rule = B2a4w/S2e3j
bo\$2o!``````
There's a lot of these floating around, so post any more you come across below!
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!

GUYTU6J
Posts: 906
Joined: August 5th, 2016, 10:27 am
Location: 中国

### Re: Smallest Linear Replicators Supporting Specific Speeds and Habits

Code: Select all

``````x = 3, y = 8, rule = B3-er4city/S23-a4city
2bo\$obo\$b2o3\$b2o\$obo\$2bo!
``````
More is in the thread "Rules with interesting replicators", including this one by 2718281828:

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``````x = 3, y = 2, rule = B2-a3ckq4ektwyz5-in6-i7e8/S1c3cijnr4ejrw5eq6ai
bo\$obo!``````
I'd like to know how these three will be classified

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``````x = 3, y = 3, rule = B2k3-qy4c/S2-n3ijkqr4ir6ae
bo\$b2o\$o!``````

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``````x = 4, y = 5, rule = B3aijn4i6n/S2-n3ijqr4ikr
3o\$o2bo\$3bo\$2b2o\$bo!``````

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``````x = 5, y = 4, rule = B2kn3-kr4iz5cr/S2-cn3ijkqr4eiqr5ckr
4bo\$2bo\$3o\$bo!``````
An oblique one:

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``````x = 5, y = 4, rule = B3aijqr4kr6n7e/S2-in3ijnqr4ir5i
3o\$obo\$o\$b2obo!
``````
Sorry but I prefer to contribute to cellular automata anonymously, so I made up a random string for my username.
-GUYTU6J