Cellular Automata in Hyperbolic Space

[Mandelbrot]

God it really has been two years since I used this account...

So recently I've been playing this game called HyperRogue, a game that takes place on a tiling of the hyperbolic plane. For those who don't know what the hyperbolic plane is, I'd recommend watching this video, and/or playing HyperRogue itself. But to provide an oversimplified explanation, hyperbolic space is a space that curves away from itself, causing lines that would be parallel in normal space to diverge in hyperbolic space.

Anyway, the grid-based nature of HyperRogue got me wondering... what if we emulated cellular automata on the hyperbolic plane?

Turns out I was not the first person to ask this question. Upon looking it up I saw someone took it upon themselves to try and put life on the hyperbolic plane, and although he didn't do much of a thorough investigation into what kind of structures or spaceships or whatever could be made using various rules and tilings, he did still get a basic program working at the end of the day. (Well, basic isn't exactly an accurate term to use given this is super complicated math and stuff but you get what I mean.)

Hopefully with this topic started you guys can look into this further, as I see this as a topic with a lot of stuff to find.

[hotdogPi]

HyperRogue actually has a cellular automata feature.

[Mandelbrot]

HyperRogue actually has a cellular automata feature.

Yes, I was aware it had that! I just made this thread so people could discuss ther findings and such.

[Mandelbrot]

Bumping this thread cause I'd love to learn more about this and this thread seems to be ignored for some reason.

Also I'm a tad lonely.

[breaker's glider gun]

Oh, living cave (and living fjord) are ca too.

[muzik]

Interestingly, "parity"/"everything replicates" rules seem to not really exist on hyperbolic grids. Plugging in the expected values just produces a pattern that explodes infinitely, likely due to the increased amount of diagonal neighbours each cell attains.



Worth noting is the fact that this is unlike the triangular Euclidean neighbourhood - it has these replication rules as you'd expect:

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 x = 1, y = 1, rule = B13579Y/S13579YL
o!

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 x = 1, y = 1, rule = B13579Y/S02468XZL
o!

[breaker's glider gun]

what app/web thing did you use? That is definetly not hyperrouge.

definetly

somehow I can't spell it right - help please!

[Ohhhhhhhhh]

definetly

somehow I can't spell it right - help please!

It’s definitely.

[breaker's glider gun]

thanks!

[Entity Valkyrie 2]

Here is the first spaceship I found on the t{3,7} hyperbolic tiling:

hyperconway 1.gif (3.22 MiB) Viewed 2401 times

EDIT: In case you were wondering what I used, I made this myself (using Python): https://github.com/ValkyRiver/hyperconway

[confocaloid]

Here is the first spaceship I found on the t{3,7} hyperbolic tiling:
hyperconway 1.gif

That's very interesting, a p4 glider on the "hyperbolic soccerball" tiling?

From the animation, it looks like the rules would be expressed 'B2am/S1ho2a' in your notation for this tiling, correct? Two birth conditions and three survival conditions. It looks like this system distinguishes between all range-1 possibilities up to rotations and reflections that preserve the tiling. That gives 18 possibilties when the middle cell is a heptagon (all neighbours are hexagons) and 20 possibilities when the middle cell is a hexagon (the neighbours alternate between hexagons and heptagons).

Did you (or anyone) consider what happens with the "outer-totalistic" approach, before going to full "isotropic" system? Can there be spaceships or other interesting reactions/objects? Simply counting alive neighbours would give only 8 possibilities when the middle cell is a heptagon, and either 7 or 16 possibilities when the middle cell is a hexagon.


Here are several related links to other pages/threads:

https://dmishin.blogspot.com/2011/10/hy ... ation.html Lightspeed glider in the unstable B2/S3 universe
https://dmishin.blogspot.com/2014/10/ce ... ic-54.html Cellular automata on hyperbolic {5,4} tiling, strobing rules
https://dmishin.github.io/hyperbolic-ca-simulator/ Hyperbolic Cellular Automata Simulator

viewtopic.php?f=7&t=2208 Hyperbolic Game Of Life?
viewtopic.php?f=11&t=594 Penrose tilings, hyperbolic tesselations
viewtopic.php?f=11&t=3893 Rules in non uniform tiling.
viewtopic.php?f=11&t=6519 other neighborhoods

[Entity Valkyrie 2]

That's very interesting, a p4 glider on the "hyperbolic soccerball" tiling?

From the animation, it looks like the rules would be expressed 'B2am/S1ho2a' in your notation for this tiling, correct? Two birth conditions and three survival conditions. It looks like this system distinguishes between all range-1 possibilities up to rotations and reflections that preserve the tiling. That gives 18 possibilties when the middle cell is a heptagon (all neighbours are hexagons) and 20 possibilities when the middle cell is a hexagon (the neighbours alternate between hexagons and heptagons).

Did you (or anyone) consider what happens with the "outer-totalistic" approach, before going to full "isotropic" system? Can there be spaceships or other interesting reactions/objects? Simply counting alive neighbours would give only 8 possibilities when the middle cell is a heptagon, and either 7 or 16 possibilities when the middle cell is a hexagon.

This was my first time working with hyperbolic cellular automata (using my self-coded Python program) — my goal was just to find a single spaceship.

As for outer-totalistic, I do know that there are 2-cell P2 oscillators in B2 rules without S0 or S1, and three-cell P2 oscillators in B2/S2. B2/S2 has explosive behavior, though.

Rules without any B1 or B2 transition (ignore B0 for now) don’t have spaceships — one such example is B4567/S3abcdhx4567, which is used in HyperRogue as the base CA for the Living Cave and Living Fjord (though these are multi-stated).

EDIT: heptagons use o/m/p/g (ortho/meta/para/gyro), similar to hexagonal neighborhood;
hexagons use x/h (only heXagons / only Heptagons), a for adjacent (b comes after a), and c for contrary (d comes after c)

EDIT 2: bug fix — the transition "4a" was mislabeled "4c".

EDIT 3: you can now import and export patterns from the clipboard

[confocaloid]

A suggestion for the script: make the heptagons (which are isolated) displayed slightly darker than hexagons (which form a "connected network"), to make it easier to distinguish cells of different shape. That could include both alive and dead cells, and could be done also for the table of formations shown on the right. (I already changed my local copy of the script to use four distinct colours, although I don't know which ones would work better.)

[...] B4567/S3abcdhx4567, which is used in HyperRogue as the base CA for the Living Cave and Living Fjord (though these are multi-stated).

Looks like that string encodes the rules "an alive cell with precisely three alive neighbours survives if and only if it's a hexagon; any cell with at least four alive neighbours lives in the next generation".

I searched, and the wiki for HyperRogue/Living Fjord claims:
"A tile with more land next to it than sea will become land, while a tile with more sea next to it than land becomes sea."
From that alone, it remains unclear whether a heptagonal tile counts as "more/less area" than a hexagonal tile. But at least it seems to say that there is symmetry between two cellstates ("land" and "sea").

Verifying the symmetry under 0/1 reversal:

• An alive hexagon remains alive in the next generation if and only if it currently has at least three alive neighbours.
  (due to presence of all survival conditions S3abcdhx456 and absence of any survival conditions S012)
• A dead hexagon remains dead in the next generation if and only if it currently has at least three dead neighbours (= at most three alive neighbours).
  (due to presence of all birth conditions B456 and absence of any birth conditions B0123abcdhx)
• An alive heptagon remains alive in the next generation if and only if it currently has at least four alive neighbours.
  (due to presence of all survival conditions S4567 and absence of any survival conditions S0123ompg)
• A dead heptagon remains dead in the next generation if and only if it currently has at least four dead neighbours (= at most three alive neighbours).
  (due to presence of all birth conditions B4567 and absence of any birth conditions B0123ompg)

So it looks like the rules expressed by 'B4567/S3abcdhx4567' are indeed symmetric under 0/1 reversal (i.e. the CA is self-complementary). But that's non-obvious (I was initially confused and thought the rules aren't 0/1-symmetric, because I expected to see some "letters" visible in the birth conditions corresponding to "letters" visible in the survival conditions). The distinction between hexagons and heptagons is partially hidden in the notation, complicating the situation.

[confocaloid]

Here is the first spaceship I found on the t{3,7} hyperbolic tiling:
hyperconway 1.gif

That's very interesting, a p4 glider on the "hyperbolic soccerball" tiling?

From the animation, it looks like the rules would be expressed 'B2am/S1ho2a' in your notation for this tiling, correct? Two birth conditions and three survival conditions. [...]

Adding the birth condition B2c ("a dead hexagon with precisely two alive neighbours, which are of different shapes and located opposite each other") enables an additional glider:

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B2acm/S1ho2a X6 X6Rb X0La X6R X6MRb X1Ra X1Rb X2 X1

two gliders in a single CA

b2acms1ho2a.png (12.69 KiB) Viewed 2318 times

edit: three different (related) spaceships in a single CA:

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B2acm3o/S1ho2a X6MRb X6M X6RLa X6MR X6MMRb X6MMR X1a X2 X1Rb X1Ra X1 X5 X4 X4Rb X4Ra

I believe the following is a p12 puffer. Try to import it from clipboard, then export back to clipboard, and compare results to see what appears to be a bug in version 1.4 (the survival condition S0a isn't correctly exported):

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B2acmx3o/S0a1ho2a3c X5RR X6LMa X6LLa X6LL X6LLLa

[CARuler]

use (press generic)

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//Automatically generated code for binary rule B 3 S 2 3
{
    //number of states
    'states': 3,

    //Neighbors sum calculation is default. Code for reference.
    //'plus': function(s,x){ return s+x; },
    //'plusInitial': 0,
    
    //Transition function. Takes current state and sum, returns new state.
    //this.generation stores current generation number
    'next': function(x, s){
        if (x===1) return 2;
        if (x===0 && (s===3)) return 1;
        if (x===2) return 0;
        return 0;
     }
}

in here with {n=3,m=7} or anything else you chose
(Ulams bones in hyperbolic plane)
edit:
spaceship

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B1h2cm3o/S02a X5a X X6

[Entity Valkyrie 2]

Adding the birth condition B2c ("a dead hexagon with precisely two alive neighbours, which are of different shapes and located opposite each other") enables an additional glider:

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B2acm/S1ho2a X6 X6Rb X0La X6R X6MRb X1Ra X1Rb X2 X1

edit: three different (related) spaceships in a single CA:

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B2acm3o/S1ho2a X6MRb X6M X6RLa X6MR X6MMRb X6MMR X1a X2 X1Rb X1Ra X1 X5 X4 X4Rb X4Ra

I believe the following is a p12 puffer. Try to import it from clipboard, then export back to clipboard, and compare results to see what appears to be a bug in version 1.4 (the survival condition S0a isn't correctly exported):

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B2acmx3o/S0a1ho2a3c X5RR X6LMa X6LLa X6LL X6LLLa

The S0 export bug has been fixed. The spaceship CARuler found is actually

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B1h2cm3o/S0o2a X5a X X6 X5Ra

So far, these spaceships all traveled along "hexagon lines" (also known as "Graveyard lines").
Here is the first spaceship that travels along a "bull line":

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B2aco/S1hx2o3c X3R X3RLa X3MRRb X3Ra X3RRa X3RL

[confocaloid]

[...] I believe the following is a p12 puffer. [...]

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B2acmx3o/S0a1ho2a3c X5RR X6LMa X6LLa X6LL X6LLLa

I think the following pattern grows forever, with population equal to (127 + t) in generation t (that is, it's a one-bit-per-tick growth).
It's not terribly satisfying, as you can't see it all at once. But the displayed population seems to match the expected value, and the way how it's constructed (twelve noninteracting p12 "hexagonal dot" puffers in different phases) means it must have the stated property.

Challenge: discover or construct a one-bit-per-tick growth with lower initial population.

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B2acmx3o/S0a1ho2a3c X0LLLLLLLLLLLLLLLLLLLLLLLa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMa X0LLR X0LLLLLLLLLLLLLLLLLLLLLLLLLLa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMa X1RRR X0LLLLLR X0LLLLLRLL X0LLLLLLLLLLLMRR X0LLLLLLLLLLLLLLLLLRL X0LLLLLLLLLLLLLLLLLLLLLLLMR X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLR X1RLRb X0LLLLLLLLLLLLLLLLMLa X0LLLLLLLLLLLLLLLLLLLLLLLMLa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLa X0LLLLLLLLLLLMa X2Ma X2Rb X1RRRRb X0LLRLLa X0LLRRb X0LLRLLMa X0LLMRRR X0LLLLLRLLLa X0LLLLLLLLMRRb X0LLLLLLLLR X0LLLLLLLLRLL X0LLLLLLLLMRRRa X0LLLLLLLLLLLMRRRb X0LLLLLLLLLLLLLLRLa X0LLLLLLLLLLLLLLM X0LLLLLLLLLLLLLLRLMa X0LLLLLLLLLLLLLLMRR X0LLLLLLLLLLLLLLLLLRLLa X0LLLLLLLLLLLLLLLLLLLLMRb X0LLLLLLLLLLLLLLLLLLLLRL X0LLLLLLLLLLLLLLLLLLLLMRRa X0LLLLLLLLLLLLLLLLLLLLLLLMRRb X0LLLLLLLLLLLLLLLLLLLLLLLLLLRa X0LLLLLLLLLLLLLLLLLLLLLLLLLLRRa X0LLLLLLLLLLLLLLLLLLLLLLLLLLMR X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLRLa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLR X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMRa X0LLLLLLLLLLLLLLLLLRb X0LLLLLLLLLLLLLLLLLLLLLLLLRb X0LLMRLRb X0LLLLLLLLRLRb X0LLLLLLLLLLLLLLMMRb X0LLLLLLLLLLLLLLLLLLLMLa X0LLLLLLLLLLLLLLLLLLLLLLLLLLMLa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLa X0LLLLLLLLLLLMRb X0LLLLLLLLLLLMRa X0LLLLLLLLLLLLRb X0LLLLLLLLLLLLLLMa X2La X3a X0LLLLLLLLLLLLLLLLLLLLLLMa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLRa X0LLRRa X0LMLa X0LLRLLRa X0LLRLMa X0LLLLLLLLMRRLa X0LLLLLLLLMRRa X0LLLLLLLLLLLLLLRLRa X0LLLLLLLLLLLLLLRRa X0LLLLLLLLLLLLLLLLLLLLMRLa X0LLLLLLLLLLLLLLLLLLLLMRa X0LLLLLLLLLLLLLLLLLLLLLLLLLLRRb X0LLLLLLLLLLLLLLLLLLLLLLLLLLRb X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMMa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLRb X1RRRb X2LLL X1RRRRa X0LLMRRRRb X0LLLLLRLLa X0LLLLLRRb X0LLLLLRLLMa X0LLLLLMRRR X0LLLLLLLLRLLLa X0LLLLLLLLLLLMRRb X0LLLLLLLLLLLR X0LLLLLLLLLLLMLa X0LLLLLLLLLLLMMa X0LLLLLLLLLLLRLL X0LLLLLLLLLLLMRRRa X0LLLLLLLLLLLLLLMRRRb X0LLLLLLLLLLLLLLLLLRLa X0LLLLLLLLLLLLLLLLLM X0LLLLLLLLLLLLLLLLLRLMa X0LLLLLLLLLLLLLLLLLMRR X0LLLLLLLLLLLLLLLLLLLLRLLa X0LLLLLLLLLLLLLLLLLLLLLLLMa X0LLLLLLLLLLLLLLLLLLLLLLLMRb X0LLLLLLLLLLLLLLLLLLLLLLLRL X0LLLLLLLLLLLLLLLLLLLLLLLMRRa X0LLLLLLLLLLLLLLLLLLLLLLLLLLMRRb X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLRa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLRRa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLMR X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLRLa X0LLLLLLLLLLLLLLLLLLLLL X0LLLLLLLLLLLLLLLLMMa X0LLLLLLLLLLLLLLLLLLLLLLLLRRb X0LLLLLLLLLLLLLLLLLLLLRb X0LLLLLLLLLLLLLLLLLLLM X0LLLLLLLLLLLLLLLLLLLLLLLLLLLR X0LLLLLLLLLLLLLLLLLLLLLLLLLLLRb X2LLRb X0LLLLLMRLRb X0LLLLLLLLLLLRLRb X0LLLLLLLLLLLLLLLLLMMRb X0LLLLLLLLLLLLLLLLLLLLLLMLa X0LLLLLLLLLLLLLLLLLLLLLLLLLLLLLMLa X0LLLLLLLLRRa X0LLLLLLLLLLLLLLMRb X0LLLLLLLLLLLLLLMRa X0LLLLLLLLLLLLLLLRb X0LLLLLLLLLLLLLLLLLLLLRa X0LLLLLLLLLLLLLLLLLMa X0LLLLLLLLLLLLLLLLMa X0LLLLLLLLLLLLLLLLLLLLLLLLRa

Also, it looks like the clipboard-importing code needs to trim any leading/trailing whitespace and newlines, otherwise copying and importing the above pattern doesn't quite work in version 1.6.

edit: here is a p2 phoenix oscillator:

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B1o2p3h/S X3Rb X4Rb X5Rb X6Rb X0Rb X1Rb X2Rb

[CARuler]

can someone tell me the the speed and direction of this ship:

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B1o2ho3d/S1h3c X X2a X0

also, some oscillators:

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B1h2p3acdghopx4achmop5opx6a7/S0ao1h2chm3ahx X X6a X5 X4

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B1h2m/S0o3a5o X X5a X0 X6a X4a

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B1h3acox4cm5p/S0ao2chm3ahx X X6a X5 X4

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B1h3acox4cm5p/S0ao2chm3ahx X5Ma X5L X5M X5LR X1 X1Ma X1M X1L

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B1h3ox/S0ao3hx X5a X X4 X5 

[confocaloid]

also, some oscillators:

FYI, the single-line export/import format very conveniently allows to put many patterns (especially many small objects of the same kind, as in this case) into the same snippet, saving vertical space for readers, like this:

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B1h2p3acdghopx4achmop5opx6a7/S0ao1h2chm3ahx X X6a X5 X4
B1h2m/S0o3a5o X X5a X0 X6a X4a
B1h3acox4cm5p/S0ao2chm3ahx X X6a X5 X4
B1h3acox4cm5p/S0ao2chm3ahx X5Ma X5L X5M X5LR X1 X1Ma X1M X1L
B1h3ox/S0ao3hx X5a X X4 X5

edit 1:

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B1o2ho3d/S1h3c X X2a X0

I believe the path of that spaceship is as follows. The spaceship is glide-symmetric, its period equals 4, its mod equals 2, and its speed (in the shown direction) is 1c/4:

B1o2ho3d/S1h3c X X2a X0

path-of-spaceship.png (29.81 KiB) Viewed 2145 times

[CARuler]

EDIT: In case you were wondering what I used, I made this myself (using Python): https://github.com/ValkyRiver/hyperconway

i have a feature request: can some short of constant play button be made? like how lifeviewer plays an animation

also, what notation can be used the represent the direction?

and what about rule tables

[Entity Valkyrie 2]

Also, it looks like the clipboard-importing code needs to trim any leading/trailing whitespace and newlines, otherwise copying and importing the above pattern doesn't quite work in version 1.6.

This has now been fixed in version 1.7, as well as an additional "history" mode.

also, what notation can be used the represent the direction?

As for directions, here are some common line patterns. HyperRogue players should know these:

hyperrogue lines.png (723.34 KiB) Viewed 2122 times

Centipede and AB lines are not present in world generation in HyperRogue, but the other three are:
• Great Wall lines are used for, well, Great Walls, which separate different lands. They are also used for lines of vines in Vineyard, mercury rivers in Terracotta Army, and the middle branch of trees in Yendorian Forest.
• Bull lines are used for the boundaries of lands in Crossroads IV, firewalls in Land of Power, lines of palace gates in Palace, ladders in Dungeon, and outer branches of trees in Yendorian Forest.
• Hexagon lines are used for the lanes in Windy Plains. Also the coasts in Warped Coast (albeit in the similar r{3, 7} tiling).

EDIT:

i have a feature request: can some short of constant play button be made? like how lifeviewer plays an animation

Automatic play has been added in version 1.8.

[confocaloid]

Here are some "controlled chaos" oscillator predecessors, a hexagon only can be born when there is an adjacent alive heptagon, a heptagon never changes its state. What are eventual periods of these?

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B2ac3abcd/S0o1o2mop3gmop4gmop5mop6o7 X3R X4R X5R X6R X0R X1R X2R X5L X5M X6L X6M X0L X0M X1L X1M X2L X2M X3L X3M X4L X4M X X4a
B2ac3abcd/S0o1o2mop3gmop4gmop5mop6o7 X3R X4R X5R X6R X0R X1R X2R X5L X5M X6L X6M X0L X0M X1L X1M X2L X2M X3L X3M X4L X4M X X3Rb
B2ac3abcd4achx5hx6a/S0o1ho2hmop3ghmop4gmop5mop6o7 X0a X0Rb X0Ma X1a X1 X1Rb X1R X2a X2Rb X2Ma X3a X3La X3Ma X3R X4a X4 X4Ra X4R X5a X5Rb X5Ma X5Ra X6a X6 X6Ma X6Ra X6R

[...] This has now been fixed in version 1.7, as well as an additional "history" mode. [...]

Confirmed (both the fix and the history mode, which helps to see directions of moving objects).

Some ideas for features:

• Maybe allow switching to a different model (band model? upper half-plane model?) where objects moving in a certain direction can keep their "apparent size" roughly the same, for example to be able to see more ash from a puffer
• Maybe allow rotating/translating view "more smoothly"? In particular, if band model or upper half-plane model is implemented, then there would be a single "special" direction in the model, and it would make sense to allow the user to choose/change what sort of a direction that is (per examples in your post).
• Autoidentify properties of low-period oscillators (up to some max. period and up to some max. population)

[LuveelVoom]

Hmm, any, uh, linear growth (is it still linear in hyperbolic space?) growth yet?

[Entity Valkyrie 2]

Hmm, any, uh, linear growth (is it still linear in hyperbolic space?) growth yet?

Yes — the first one found was this P12 puffer by confocaloid:

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B2acmx3o/S0a1ho2a3c X5RR X6LMa X6LLa X6LL X6LLLa

EDIT:

Here are some "controlled chaos" oscillator predecessors, a hexagon only can be born when there is an adjacent alive heptagon, a heptagon never changes its state. What are eventual periods of these?

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B2ac3abcd/S0o1o2mop3gmop4gmop5mop6o7 X3R X4R X5R X6R X0R X1R X2R X5L X5M X6L X6M X0L X0M X1L X1M X2L X2M X3L X3M X4L X4M X X4a
B2ac3abcd/S0o1o2mop3gmop4gmop5mop6o7 X3R X4R X5R X6R X0R X1R X2R X5L X5M X6L X6M X0L X0M X1L X1M X2L X2M X3L X3M X4L X4M X X3Rb
B2ac3abcd4achx5hx6a/S0o1ho2hmop3ghmop4gmop5mop6o7 X0a X0Rb X0Ma X1a X1 X1Rb X1R X2a X2Rb X2Ma X3a X3La X3Ma X3R X4a X4 X4Ra X4R X5a X5Rb X5Ma X5Ra X6a X6 X6Ma X6Ra X6R

Their periods become 24, 4, and 36 respectively.

EDIT 2: version 1.9 now has a "detect period" feature.

EDIT 3:

can someone tell me the the speed and direction of this ship:

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B1o2ho3d/S1h3c X X2a X0

I found two P16 spaceships in your rule:

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B1o2ho3d/S1h3c X0a X0 X6R X5 X5a X4a X4Ma X4La X4RRb X5LLa
B1o2ho3d/S1h3c X0 X2a X1 X0MRa X0MM X0MMR X3La X3Ma X2RLRa X2RLMa X3LLMa X3LLLRb

I also found a P36 spaceship:

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B1o2ho3d/S1h3c X4a X3Rb X3a X4Ma X4Ra X4MRa X4MRb X3La X2Ra X5Ma X5LRb X6Rb X6Ra X6MRb X6RRb X0Ma

[confocaloid]

A somewhat interesting oscillator, 21P288 (period 288, minimum population 21):

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B2ac3abcd4achx5hx6a/S0o1ho2hmop3ghmop4gmop5mop6o7 X0 X0R X1R X2 X3 X4 X5 X6R X0La X0Ma X1a X0RRa X1RLa X2Ma X2Ra X2a X3La X4a X4La X5a X5La

B2ac3abcd4achx5hx6a/S0o1ho2hmop3ghmop4gmop5mop6o7 X0 X0R X1R X2 X3 X4 X5 X6R X0La X0Ma X1a X0RRa X1RLa X2Ma X2Ra X2a X3La X4a X4La X5a X5La

21P288-b2ac3abcd4achx5hx6as0o1ho2hmop3ghmop4gmop5mop6o7.png (26.85 KiB) Viewed 2008 times

edit 2025-02-21:

As for directions, here are some common line patterns. HyperRogue players should know these:
hyperrogue lines.png

I decided to draw those directions as patterns and export them:

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B/S X0a X1a X X4a X3Rb X3R X3RRa X3RLa X0 X0Ma X0LRb X0LR X0LRLa X0LRRa X0LRL X0LRLMa X0LRLLRb X0LRLLR X0LRLLRLa X0LRLLRRa X0LRLLRL X3RL X3RLMa X3RLLRb X3RLLR X3RLLRRa X3RLLRLa X3RLLRL

B/S X X1a X2a X1 X1Rb X1Ra X1R X1MRb X1RLa X1MR X0a X6a X6 X5Rb X6La X5R X5RRb X5RRa X5RR X5RLRb X5RRLa X5RLR X1MRRb X1MRRa X1MRR X5RLRRb X5RLRRa X5RLRR X5RLRLRb X5RLRRLa X5RLRLR X1MRLRb X1MRRLa X1MRLR X1MRLRRb X1MRLRRa X1MRLRR

B/S X X2a X1Rb X1R X6a X5Rb X5R X5RLa X5MRRb X5MRR X1RRa X1RLR X1RLRb X1RLRLa X1RLMRRb X1RLMRR X1RLMRRRa X1RLMRRLRb X1RLMRRLR X5MRRRa X5MRRLRb X5MRRLR X5MRRLRLa X5MRRLMRRb X5MRRLMRR

B/S X5a X4a X3a X2Rb X2Ra X2MRb X4Rb X5La X4RRb X2MRa X2MMRb X2MMRa X2MMMRb X5LLa X4RRRb X5LLLa X4RRRRb X4 X3 X2R X2MR X4R X4RR X X5 X2 X2M X5L X5LL X4RRR X5LLL X2MMR X2MM X2MMM X3Rb X6a X2a X3La X4Ra X5Ma X2Ma X2RLa X4RRa X2MMa X2MRLa X2MMMa X5LMa X4RRRa X5LLMa

B/S X X6a X5a X2a X3a X2 X2La X2Ma X2L X2LMa X2LRa X2LM X5 X5Ra X5Ma X5M X5MMa X5MLa X5ML X5MLRa X5MLMa X5MLM X2LMMa X2LMLa X2LML