Cellular Automata in Hyperbolic Space

[Entity Valkyrie 2]

Funny P35:

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B2achmp4c5mo/S1o2aox3abdmp4chmp5mo6o X6a X5a X X2a X3a X4a X0a X0

I also looked at this rule, which has a P6 bull-line traveling spaceship:

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B2aco/S1hx2op3c X X3Rb X4a X6a X3a X3

It apparently has a P70:

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B2aco/S1hx2op3c X3 X X3Rb X2a X2Rb

B2acoS1hx2op3c X3 X X3Rb X2a X2Rb.png (425.47 KiB) Viewed 2110 times

[confocaloid]

Examples of an exponential population growth:

• population sequence 1, 8, 22, 50, 99, 183, 330, 582, 1016, 1765, 3053, 5272, ... (related: OEIS A290398)

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  B1234567/S01234567 X

• population sequence 1, 7, 22, 49, 97, 181, 325, 574, 1003, 1741, ...

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  B1234567/S01234567 X4a

• population sequence 1, 4, 10, 22, 40, 70, 115, 187, 298, 472, 742, 1162, 1816, 2833, ...

  Code: Select all

  B1hx2achx3abcdhx4achx5hx6a/S01234567 X4a

How slow can be an exponential population growth (on this tiling, and for this type of cellular automata)? How sparse can be the resulting pattern? Which population sequences are possible starting from just a single alive heptagon, or from a single alive hexagon?

muzik wrote:
Would exponential growth be possible with a linearly expanding neighbourhood?

The most straightforward way to get exponential growth would be to work on a regular tiling of a hyperbolic surface (where the angles of a triangle sum to less than 180 degrees). For example, such a surface could support a tiling of regular octagons, where each cell has eight neighbours, but the number of cells at distances of 1,2,3... increases exponentially. See https://en.wikipedia.org/wiki/Uniform_t ... olic_plane.

[Entity Valkyrie 2]

A potential omniperiodic rule?

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B2c3bdho4ac5o7/S0o12acmox3ac4a X0LM X0ML X0LMa X6MR X6ML X6LM X6LMa X6LLa X5RR X5RRa X5RLa X5MR X5MRa X5MMa X5ML X5LM X5LMa X5LLa X4RR X4RRa X4RLa X4MR X4MRa X4MMa X4ML X4LM X4LMa X4LLa X3RR X3RRa X3RLa X3MR X3MRa X3MMa X3ML X3LM X3LMa X3LLa X2RR X2RRa X2RLa X2MR X2MRa X2MMa X2ML X2LM X2LMa X2LLa X1RR X1RRa X1RLa X1MR X1MRa X1MMa X1ML X1LM X1LMa X1LLa X0RLa X0MR X0MMa X0MRa X0RRa X0RR X6RR X6RRa X6LRRb X6LRRa X5LRRb X5LRRa X4LRRb X4LRRa X3LRRb X3LRRa X2LRRb X2LRRa X1LRRb X1LRRa X0LRRb X0LRRa X6MRa X0LLa X0L X0LRa X6MLa

potentially omniperiodic rule hyperconway.png (383.88 KiB) Viewed 2097 times

This signal is made up of hypercyclic segments that go 7-6-6-7-6-6-7, where every heptagon has a 2/7-turn corner. The "reflectors" form 1/7-turn corners. Two signals can follow each other 5 ticks apart, and periods 1−4 exist in this rule:

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B2c3bdho4ac5o7/S0o12acmox3ac4a X0Rb X1La X4R X4M X4MR X3LRb X3LL X3LMa X3LRa X6M X0Ra X0MRb X0RLa X0RRa X0RRb X3LR X3LLa X2R X3

Though this might not be sufficient to prove omniperiodicity, since I don't know of a way to show whether it is possible to form a loop with any period.

[CARuler]

Exponential Growth!!!

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B2ampx3m/S0a3c X3a X0 X1a

[confocaloid]

Exponential Growth!!!

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B2ampx3m/S0a3c X3a X0 X1a

Looks like that's a replicator? Or at the very least a failed replicator? Compare generation 17 (a "hexagonal dot" and a 9-bit symmetric configuration) vs. generation 25 (the same "hexagonal dot" and two instances of the same 9-bit symmetric configuration).

The population sequence (beginning in generation 0) is as follows, beginning at the fourth row every row appears to be (2 x (previous row) - 1):

3, 5, 5, 6, 6, 7, 7, 10,
4, 7, 7, 7, 9, 8, 7, 9,
5, 10, 7, 11, 14, 11, 13, 18,
9, 19, 13, 21, 27, 21, 25, 35,
17, 37, 25, 41, 53, 41, 49, 69,
33, 73, 49, 81, 105, 81, 97, 137,
65, 145, 97, 161, 209, 161, 193, 273,
129, 289, 193, 321, 417, 321, 385, 545,
257, 577, 385, 641, 833, 641, 769, 1089,
513, 1153, 769, 1281, 1665, 1281, 1537, 2177,
1025, 2305, 1537, 2561, 3329, 2561, 3073, 4353,
2049, 4609, 3073, 5121, 6657, 5121, 6145, 8705,
4097, 9217, 6145, 10241, 13313, 10241, 12289, 17409,
...

B2ampx3m/S0a3c X3a X0 X1a

comparison-between-gen17-and-gen25.png (79 KiB) Viewed 2039 times

[CARuler]

Exponential Growth!!!

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B2ampx3m/S0a3c X3a X0 X1a

wait... then what is the growth of this??? (not quadratic)

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B12amop3gmop/S01o2mop3abgmop4ao6a7 X

edit: there are different types of strobing rules!!!!

• B0/S (p2)
• B0o/S (p2)
• B0a/S (p2)
• B0/S6a (p3)
• B0/S7 (p3)
• B07/S6a (p4)
• B03h/S7 (p4)
• B0a7/S (p3)
• B0a7/S6x (p4)
• B0o3h/S (p3)
• B0o3h/S0o (p3)
• B0o3h/S0o6a (p4)

is that all?

[confocaloid]

[...] wait... then what is the growth of this??? (not quadratic)

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B12amop3gmop/S01o2mop3abgmop4ao6a7 X

[...]

Looks like that is the same evolutionary sequence as in B1234567/S01234567, it is also exponential:

[...] population sequence 1, 8, 22, 50, 99, 183, 330, 582, 1016, 1765, 3053, 5272, ... (related: OEIS A290398)

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B1234567/S01234567 X

[...]

[CARuler]

i have created a ruletable notation for t{3,7}

for n_states different states can be available for heptagons and hexagons (using (hex:n, hept:m)); similarly, certain variables can only be used in one or the other, by defalt "var x = list" acts for all shapes, "var hex x = list" acts for only hexagons and "var hept x = list" acts only for heptagons; for transitions "hex transition" means the center cell is a hexagon (the first state after the center is on a hexagon), and "hept transition" means the center cell is a heptagon; for colors "hex state color" means the has the state has that color while on a hexagon and "hept color color" is the same but with a heptagon.

demo:
a ship traveling across the great wall:

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Rule=hyperbolicShip X X3a X5a
@RULE hyperbolicShip

@TABLE
n_states:(hex:3,hept:3)
neighborhood:Moore(t{3,7})
symmetries:rotateAllReflect
var a = {0,1,2}
var b = a
var c = a
var d = a
var e = a
var f = a
var hept g = a

hex 0,0,1,0,0,0,0,2
hept 1,1,0,1,0,0,0,0,1
hept 0,2,2,0,0,0,0,0,1
hex 0,2,0,0,0,0,0,1
hex 0,1,1,1,0,0,0,1
hex 0,1,1,0,0,0,0,1
hept 0,1,2,0,0,0,0,0,2
hex 0,1,2,0,0,0,0,1
hex 1,1,0,0,0,0,0,1
hept 0,1,1,1,0,0,0,0,1
hex 1,0,2,0,2,0,0,1

#defaults
hex 1,a,b,c,d,e,f,0
hex 2,a,b,c,d,e,f,0
hept 1,a,b,c,d,e,f,g,0
hept 2,a,b,c,d,e,f,g,1
@COLORS
hex 0 255 255 255
hept 0 192 192 192
hex 1   0 255 255
hept 1   0 128 128
hex 2 255 128   0
hept 2 128  64   0

edit:
can strobing rules be implemented after this?
edit2:
on the other simulator
*had these lying about*

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https://dmishin.github.io/hyperbolic-ca-simulator/index.html
{7,3}


7$3$|1(a3(B|3))(a(B|1))


7$3$(a(B(a2(B(a2(B(A3(b(A3(b(A2(b(A3(B(a(B(a2(B(A3(B|1(a(B|3))))(a3(B|2))))))))))))))))))(A2(b(A3(B(a(B(A3(B(a2(B(a3(B|2))))))(A2(b(A3(B(a(B|1(a2(B|3))))))))))))))))(B(A2(b(A3(B(a(B(A2(b(A3(B(a(B(A3(B(a(B(A3(B(a(B|1))(a2(B|3))))(a3(B|1))))))))))))))))))(a2(B(a2(B(A3(b(A3(B(a(B(a2(B(A3(B(a(B(A3(B(a(B|1))(a2(B|3))))(a3(B|1))))))))))))))(a3(B(a2(B(a2(B(a3(B(a2(B|1))))(a2(B(A3(B(a2(B|3))))(A2(b(A3(B|1)))))))))))))))))(b(A2(b(A3(B(a(B(A3(B(a(B(A2(b(A3(B(a(B|3(A3(B(a(B|1(A3(B|3))))(a2(B|2))))))))))))))))))))(A3(B(a2(B(a3(B(a2(B(a2(B(A3(B(a(B(A3(B(a(B(a3(B|1))(A3(B(a2(B|3))(a(B|1))))))))))))))))))))(a(B(A3(B(a(B(A3(B(a(B(a2(B(A3(B(a(B(a3(B|1))(A3(B(a2(B|3))(a(B|1)))))))))))))))))))))))(A(B(a(B(a2(B(a3(B(a2(B(a2(B(a2(B(a2(B(A3(B(a(B(A3(B|1(a(B|3))))(a3(B|2))))))))))))))))))))(a2(B(a2(B(A3(b(A3(B(a2(B(a2(B(a2(B(A3(B(a(B(a3(B|3))(A3(B|1))))(a2(B|1))))))))))))))))))))(a2(B(a2(B(A3(B(a(B(A3(B(a(B(A3(B(a(B(A3(B(a(B(a3(B|3))(A3(B|1))))(a2(B|1))))))))))))))))))))


A=
//Automatically generated code for binary rule B 3 S 2 3
{
    //number of states
    'states': 4,

    //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 && (s===4)) return 1;
        if (x===3 && (s===1)) return 2;
        if (x===0 && (s===4)) return 3;
        if (x===3 && (s===3)) return 1;
        if (x===2 && (s===4)) return 1;
        return 0;
     }
}

B=
//Automatically generated code for binary rule B 3 S 2 3
{
    //number of states
    'states': 4,

    //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 && (s===4 || s===0)) return 1;
        if (x===3 && (s===1)) return 2;
        if (x===0 && (s===4)) return 3;
        if (x===3 && (s===3)) return 1;
        if (x===2 && (s===4)) return 1;
        return 0;
     }
}

A{30,3}

30$3$(B(a8(B(A15(B(a(B(A3(B(a(B(A3(B|3(a(B|3(A4(B|1))(A3(B(a(B|1))(a2(B|3))))))(a2(B|1(A3(B|3))))(a3(B|2))))(A4(B|3(a(B(A3(B|3))))(a2(B|2))))))))))))(a14(B(a2(B(A3(B(a(B(A3(B(a2(B|2))(a(B|1(A3(B|3))))))))))))))(a12(B(A4(B(a(B(A3(B(a(B(A3(B|1(a(B|3))))(A4(B|2))))))))))))(a11(B(A3(B(a(B(A3(B(a(B|3(A4(B|1(a(B|3))))(A3(B|3(a(B(A4(B|3))(A3(B|1))))(a2(B|1))))(A5(B|2))))(a2(B|3(A3(B(a(B|3))))(A4(B|2))))))))))))))(a9(B(a5(B(A2(b(A3(B(a(B(A3(B(a(B(A3(B(a2(B|3))(a(B|1))))(A4(B|1))))))))))))))(A4(B(a(B(A3(B(a(B(A3(B(a(B(A3(B(a2(B|3))(a(B|1))))(A4(B|1))))))))))))))))(A10(B(a13(B(A2(b(A3(B(a(B(A3(B(a(B(A3(B(a(B|1))(a2(B|3))))(A4(B|1)))))))))))))))))(a14(B(a7(B(A2(b(A3(B(a(B(A3(B(a(B(A3(B(a(B|1))(a2(B|3))))(A4(B|1))))))))))))))))(A(B(a11(B(A9(B(a(B(a2(B(A3(B(a(B(A3(B(a2(B|1))(a(B(A4(B|3))(A3(B|1))))))))))))))))))(a7(B(A14(B(a(B(A3(B(a(B(A3(B(a(B(A3(B(a(B|1))(a2(B|3))))(A4(B|1))))))))))))))))))(A5(B(a6(B(A3(B(a(B(A3(B(a(B(A3(B(a(B(A3(B|1(a(B|3))))(A4(B|2))))))))))))))))))(A8(b(A3(B(a(B(a2(B(A3(B(a(B(A3(B(a(B(A4(B|3))(A3(B|1))))(a2(B|1))))))))))))))))

7$3$|1(A2(B|2))(A3(B|3))(B|1(a2(B|3)))




C=

//Automatically generated code for binary rule B 3 S 2 3
{
    //number of states
    'states': 7,

    //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===0 && s===5) return 6;
        if (x===3 && s===3) return 1;
        if (x===0 && s===4) return 1;
        if (x===1 && s===21) return 1;
        if (x===6 && s===13) return 6;
        if (x===6 && s===14) return 4;
        if (x===6 && s===6) return 1;
        if (x===6 && s===8) return 3;
        if (x===0 && s===14) return 2;
        
        return 0;
     }
}

4$5$|1(A2(B|3))(A(b2|2))


D=
//Automatically generated code for binary rule B 3 S 2 3
{
    //number of states
    'states': 7,

    //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===0 && s===5) return 6;
        if (x===3 && s===3) return 1;
        if (x===0 && s===4) return 1;
        if (x===1 && s===21) return 1;
        if (x===6 && s===13) return 6;
        if (x===6 && s===14) return 4;
        if (x===6 && s===6) return 1;
        if (x===6 && s===8) return 3;
        if (x===0 && s===14) return 2;
        if (x===0 && s===72) return 5;
        if (x===4 && s===0) return 6;
        if (x===6 && s===12) return 6;
        if (x===6 && s===0) return 4;
        
        return 0;
     }
}

4$5$|1(A(B2(A2(B2(A2(B|4)))))(b2|2))(A2(B|3))(B2(a(B2(A2(B|5)))))


4$5$|6(b(A(B2|1(a(B2|6(A2(b2|1)(B2|1)(B|1))(a(B|1)))(B|1(a(B2|1)(B|1)))))(b2|1(A(b|1(A(b|1)(b2|1)))))))(A2(B|6))(A(b2|6))

4$5$|6(b(A(B2|6)(b|6)(b2|6)))(B2|6)(B|6)

4$5$|6(b2|6)(B(A2(b(A(b(A(b2|6)))))))(B2|6)

5$4$(b|4)(a2(B|4))

[R2INT]

To me, a more viable notation for potential ruletables using this particular hexagon-heptagon tessellation is as follows:

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@RULE Hex-Hept-Test
@COLORS
0x 0,0,0
0p 32,32,32
1x 255,255,255
1p 224,224,224
2x 0,255,255
2p 32,224,224
3x 255,0,255
3p 224,32,224
4x 255,255,0
4p 224,224,32
5x 0,0,255
5p 32,32,224
6x 0,255,0
6p 32,224,32
7x 255,0,0
7p 224,32,32
@TABLE
n_states:8
neighborhood:Hyperbolic667Moore
symmetries:rotate67reflect
0 1,1,0,0,0,0 1 # applies to hexagons only
0 1,2,1,0,0,0,0 2 # applies to heptagons only

Based on the symmetries, we have:

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none (anisotropic)
permute (OT)
reflectx
reflectp
reflect
rotate2
rotate2reflectx
rotate2reflectp
rotate2reflect
rotate3
rotate3reflectx
rotate3reflectp
rotate3reflect
rotate6
rotate6reflectx
rotate6reflectp
rotate6reflect
rotate7
rotate7reflectx
rotate7reflectp
rotate7reflect
rotate27
rotate27reflectx
rotate27reflectp
rotate27reflect
rotate37
rotate37reflectx
rotate37reflectp
rotate37reflect
rotate47
rotate47reflectx
rotate47reflectp
rotate47reflect
rotate67
rotate67reflectx
rotate67reflectp
rotate67reflect (isotropic)

Here, if the digit 7 is in the symmetries, this means to rotate in the heptagons only.

Reflectx only reflects the hexagons, reflectp reflects only the heptagons, and reflect reflects both the hexagons and the heptagons.

Attempting to translate CARuler's HyperbolicShip:

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@RULE hyperbolicShip
@COLORS
0x 0,0,0
0p 32,32,32
1x 255,255,255
1p 224,224,224
2x 0,255,255
2p 32,224,224
@TABLE
n_states:3
neighborhood:Hyperbolic667Moore
symmetries:rotate67reflect
var a = {0,1,2}
var b = a
var c = a
var d = a
var e = a
var f = a
var g = a
0 0,1,0,0,0,0 2
1 1,0,1,0,0,0,0 1
0 2,2,0,0,0,0,0 1
0 2,0,0,0,0,0 1
0 1,1,1,0,0,0 1
0 1,1,0,0,0,0 1
0 1,2,0,0,0,0,0 2
0 1,2,0,0,0,0 1
1 1,0,0,0,0,0 1
0 1,1,1,0,0,0,0 1
1 0,2,0,2,0,0 1

[confocaloid]

Based on the symmetries, we have:

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none (anisotropic)

The "none" choice doesn't really make sense on a hyperbolic plane. The reason is that a rotation can be decomposed into translations. You can do several consecutive translations "along" different hyperbolic lines, and end up in the original location where you were at the beginning except rotated by a nonzero angle.

This is unlike the (Euclidean) plane, where any sequence of consecutive translations always makes a translation and never makes a rotation or reflection.

Note also that a closely related similar problem would happen if one tried to define "symmetries:none" on a spherical tiling (such as the 20 spherical triangles projected from faces of an inscribed icosahedron).

A consequence is that arbitrary non-isotropic cellular automata on uniform tilings can only make sense in the Euclidean case. (Even in the Euclidean case, non-isotropic CA feel much more arbitrary and less "natural", for a reasonable common-sense meaning of 'natural'.)

[R2INT]

The "none" choice doesn't really make sense on a hyperbolic plane[...]

Good point! Here is a revised version of symmetry listings, taking into account the "holonomy" that results when moving across cells:

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permute (OT)
rotate67 (minimum symmetry)
rotate67reflectx (reflect on hexagons only)
rotate67reflectp (reflect on heptagons only)
rotate67reflect (isotropic)

[confocaloid]

[...] Here is a revised version of symmetry listings, taking into account the "holonomy" that results when moving across cells:

Code: Select all

permute (OT)
rotate67 (minimum symmetry)
rotate67reflectx (reflect on hexagons only)
rotate67reflectp (reflect on heptagons only)
rotate67reflect (isotropic)

The "rotate6" prefix looks dubious, shouldn't it be "rotate3" instead? The tiles surrounding a hexagon alternate in shape (hexagon, heptagon, hexagon, heptagon, hexagon, heptagon) so there are only three rotations instead of six.

For example, in the notation described earlier and used in the existing simulator above, B3x and B3h are two different rules "matching" two different cell conditions.
(Someone might want to introduce "pseudo-symmetries" named "rotate6...", where those two cell conditions are "merged" into a single condition, like how on the square tiling one could "merge" B4c with B4e and "merge" S2i with S2n via cyclic permutations. But such cyclic permutations don't correspond to actual geometric symmetries of the tiling.)

[R2INT]

[...]
The "rotate6" prefix looks dubious, shouldn't it be "rotate3" instead? The tiles surrounding a hexagon alternate in shape (hexagon, heptagon, hexagon, heptagon, hexagon, heptagon) so there are only three rotations instead of six.
[...]

You're right! I completely missed the 3-fold symmetry of the hexagons. Let me revise the list of symmetries:

Code: Select all

permute (OT)
rotate37 (minimum symmetry)
rotate37reflectx (reflect on hexagons only)
rotate37reflectp (reflect on heptagons only)
rotate37reflect (isotropic)
permute (OT)
rotate67 (psuedo symmetry)
rotate67reflectx
rotate67reflectp
rotate67reflect

Are there any further revisions necessary?

[eRroR_6o6]

Theoretically, there could be a 2-cell spaceship, but for now, have this 3-cell minimalistic p2 "photon".

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B1o2amx/S0a X X5 X4a

[confocaloid]

[...]
The "rotate6" prefix looks dubious, shouldn't it be "rotate3" instead? The tiles surrounding a hexagon alternate in shape (hexagon, heptagon, hexagon, heptagon, hexagon, heptagon) so there are only three rotations instead of six.
[...]

You're right! I completely missed the 3-fold symmetry of the hexagons. Let me revise the list of symmetries:

Code: Select all

permute (OT)
rotate37 (minimum symmetry)
rotate37reflectx (reflect on hexagons only)
rotate37reflectp (reflect on heptagons only)
rotate37reflect (isotropic)
permute (OT)
rotate67 (psuedo symmetry)
rotate67reflectx
rotate67reflectp
rotate67reflect

Are there any further revisions necessary?

I think it's an open question whether any further revisions are necessary.
However, there is much more flexibility that can be added into the design, in a consistent way.

On this tiling ("hyperbolic soccerball") and with this neighbourhood (the neighbours are the side-by-side adjacent cells), the rules can be split into two sets: one set of rules telling how a heptagonal cell "reacts to its environment", and another set of rules telling how a hexagonal cell "reacts to its environment".

There is no hard reason for disallowing two completely independent choices of symmetry types, one for hexagons, another for heptagons.

There is also no hard reason for disallowing different sets of cellstates. A heptagonal cell can never become a hexagonal cell during evolution. A hexagonal cell also can never become a heptagonal cell. So one can have two separate sets of cellstates; both can be numbered from zero, but for example "a state-1 heptagon" and "a state-1 hexagon" describe cells in two different cellstates (despite both numbered 1).

There could be H distinct states (0 through H-1) for heptagons, and X distinct states (0 through X-1) for hexagons. (An exception is that, if one allows arbitrary "rotate6" cyclic shifts that don't correspond to geometric symmetries, then H = X appears to be necessary and there is only one set of cellstates instead of two. Otherwise, H can be different from X.)

For heptagons, one can at least choose between the following:

• "permute" (all permutations of neighbours). With two cellstates, that leads to eight heptagonal cell conditions, from 0 through 7 alive neighbours.
• "rotate7" (arbitrary cyclic shifts). With two cellstates. that leads to 20 heptagonal cell conditions.
• "rotate7reflect" (arbitrary cyclic shifts and reversals). With two cellstates, that leads to 18 heptagonal cell conditions.

For hexagons, one can at least choose between

• "permute" (all permutations of neighbours). With two cellstates, that leads to seven hexagonal cell conditions, from 0 through 6 alive neighbours.
• "permute_hx" (the three hexagonal neighbours are permuted separately, and the three heptagonal neighbours are also permuted separately). With two cellstates, that leads to 16 hexagonal cell conditions (0 through 3 alive hexagonal neighbours and 0 through 3 alive heptagonal neighbours).
• "rotate3" (cyclic shifts that correspond to geometric symmetries of the tiling). With two cellstates, that leads to 24 hexagonal cell conditions.
• "rotate3reflect" (reversals and cyclic shifts that correspond to geometric symmetries of the tiling). With two cellstates, that leads to 20 hexagonal cell conditions.
• "rotate6" (arbitrary cyclic shifts). With two cellstates, that leads to 14 hexagonal cell conditions.
• "rotate6reflect" (arbitrary cyclic shifts and reversals). With two cellstates, that leads to 13 hexagonal cell conditions.

The above already gives at least 3 x 6 = 18 possible choices for the equivalence relation that determines (for a given choice of H, X) what are all the distinguishable cell conditions.

The existing simulator for two-state CA (https://github.com/ValkyRiver/hyperconway) uses what is equivalent to "rotate7reflect" for heptagons and "rotate3reflect" for hexagons, with two cellstates in both cases. This leads to the space of 2^18 x 2^20 = 2^38 CA (if B0 is permitted).

[CARuler]

you could add symmetries:none if each state had a rotated version, like langton's ant or an arrow for each state

[unname4798]

you could add symmetries:none if each state had a rotated version, like langton's ant or an arrow for each state

No condition required.™

[confocaloid]

you could add symmetries:none if each state had a rotated version, like langton's ant or an arrow for each state

No condition required.™

In general "none" is no longer possible to define consistently while accounting for translations, because a rotation can be decomposed into finitely many translations.

There may be special cases/restrictions such that one can consistently define some sort of non-isotropic CA on the "hyperbolic soccerball" tiling. But it remains to be explained whether/how that could be done.

One possibility is CA where rotations don't change evolution of patterns, but reflections can change evolution of patterns. (In this case, a reflection cannot be decomposed into finitely many translations.) Then every asymmetric cell condition would turn into two different cell conditions (mirror images of each other).

[...] a rotation can be decomposed into translations. You can do several consecutive translations "along" different hyperbolic lines, and end up in the original location where you were at the beginning except rotated by a nonzero angle. [...]

[CARuler]

Or one where when you rotate the pattern, you change the cell states

[CARuler]

anther ship

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B1h2m3d/S0o1o3d6o X X5a X6

edit:
a path exponential growths can go down

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B/S X3Rb X4a X X0a X6Rb X1a X0Rb X6R X0R X6RRa X6RLRb X6RLa X6MRRb X0RLa X0MRRb X0RRa X0RLRb X0RLR X0MRR X0MRRLa X0MRLRRb X0MRRRa X0MRRLRb X0RLRLa X0RLMRRb X0RLRRa X0RLRLRb X6MRR X6MRRRa X6MRRLRb X6MRRLa X6MRLRRb X6RLR X6RLRLa X6RLMRRb X6RLRRa X6RLRLRb X6MRRLR X6MRRLRLa X6MRRLMRRb X6MRRLRRa X6MRRLRLRb X6MRLRR X6MRLRRRa X6MRLRRLRb X6MRLRRLa X6MRLRLRRb X6RLMRR X6RLMRRLa X6RLMRLRRb X6RLMRRRa X6RLMRRLRb X6RLRLR X6RLRLRLa X6RLRLMRRb X6RLRLRRa X6RLRLRLRb X0MRLRR X0MRLRRLa X0MRLRLRRb X0MRLRRRa X0MRLRRLRb X0MRRLR X0MRRLRLa X0MRRLMRRb X0MRRLRRa X0MRRLRLRb X0RLMRR X0RLMRRLa X0RLMRLRRb X0RLMRRRa X0RLMRRLRb X0RLRLR X0RLRLRRa X0RLRLRLRb X0RLRLRLa X0RLRLMRRb

edit2:
a rake!!!!!!!!!

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B1o2amx/S0a X X0La X0Ra X0R X0LRb X0RLa X1La X1Ma X1M X1RRb X2 X2Rb X2Ma X2L X2M X2RRb X3a X3La X3LRb X3RRa X3RRb X4a X4Ma X4L X4RLa X4RRb X5 X5R X5M X5RLa X5RRa X5RRb X6Rb X6La X6Ma X6Ra X6LRb X6MRb X6RLa X6RRa

the rule:

Theoretically, there could be a 2-cell spaceship, but for now, have this 3-cell minimalistic p2 "photon".

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B1o2amx/S0a X X5 X4a

edit3:

Exponential Growth!!!

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B2ampx3m/S0a3c X3a X0 X1a

the path:

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B2ampx3m/S0a3c X6MLMMMMMMMMMMMMMMa X6MLMMMMMMMMMMMMMMRb X6MLMMMMMMMMMMMMMRRb X6MLMMMMMMMMMMMMRRa X6MLMMMMMMMMMMMRRLa X6MLMMMMMMMMMMMRRLRb X6MLMMMMMMMMMMMRRRRb X6MLMMMMMMMMMMRLLMa X6MLMMMMMMMMMRRLMMa X6MLMMMMMMMMMRRLMMRb X6MLMMMMMMMMMRRLMRRb X6MLMMMMMMMMMRRLRRa X6MLMMMMMMMMMRRRRLa X6MLMMMMMMMMMRRRRLRb X6MLMMMMMMMMMRRRRRRb X6MLMMMMMMMMRLLLLMa X6MLMMMMMMMRRLMMMMa X6MLMMMMMMMRRLMMMMRb X6MLMMMMMMMRRLMMMRRb X6MLMMMMMMMRRLMMRRa X6MLMMMMMMMRRLMRRLa X6MLMMMMMMMRRLMRRLRb X6MLMMMMMMMRRLMRRRRb X6MLMMMMMMMRRLRLLMa X6MLMMMMMMMRRRRLMMa X6MLMMMMMMMRRRRLMMRb X6MLMMMMMMMRRRRLMRRb X6MLMMMMMMMRRRRLRRa X6MLMMMMMMMRRRRRRLa X6MLMMMMMMMRRRRRRLRb X6MLMMMMMMMRRRRRRRRb X6MLMMMMMMRLLLLLLMa X6MLMMMMMRRLMMMMMMa X6MLMMMMMRRLMMMMMMRb X6MLMMMMMRRLMMMMMRRb X6MLMMMMMRRLMMMMRRa X6MLMMMMMRRLMMMRRLa X6MLMMMMMRRLMMMRRLRb X6MLMMMMMRRLMMMRRRRb X6MLMMMMMRRLMMRLLMa X6MLMMMMMRRLMRRLMMa X6MLMMMMMRRLMRRLMMRb X6MLMMMMMRRLMRRLMRRb X6MLMMMMMRRLMRRLRRa X6MLMMMMMRRLMRRRRLa X6MLMMMMMRRLMRRRRLRb X6MLMMMMMRRLMRRRRRRb X6MLMMMMMRRLRLLLLMa X6MLMMMMMRRRRLMMMMa X6MLMMMMMRRRRLMMMMRb X6MLMMMMMRRRRLMMMRRb X6MLMMMMMRRRRLMMRRa X6MLMMMMMRRRRLMRRLa X6MLMMMMMRRRRLMRRLRb X6MLMMMMMRRRRLMRRRRb X6MLMMMMMRRRRLRLLMa X6MLMMMMMRRRRRRLMMa X6MLMMMMMRRRRRRLMMRb X6MLMMMMMRRRRRRLMRRb X6MLMMMMMRRRRRRLRRa X6MLMMMMMRRRRRRRRLa X6MLMMMMMRRRRRRRRLRb X6MLMMMMMRRRRRRRRRRb X6MLMMMMRLLLLLLLLMa X6MLMMMRRLMMMMMMMMa X6MLMMMRRLMMMMMMMMRb X6MLMMMRRLMMMMMMMRRb X6MLMMMRRLMMMMMMRRa X6MLMMMRRLMMMMMRRLa X6MLMMMRRLMMMMMRRLRb X6MLMMMRRLMMMMMRRRRb X6MLMMMRRLMMMMRLLMa X6MLMMMRRLMMMRRLMMa X6MLMMMRRLMMMRRLMMRb X6MLMMMRRLMMMRRLMRRb X6MLMMMRRLMMMRRLRRa X6MLMMMRRLMMMRRRRLa X6MLMMMRRLMMMRRRRLRb X6MLMMMRRLMMMRRRRRRb X6MLMMMRRLMMRLLLLMa X6MLMMMRRLMRRLMMMMa X6MLMMMRRLMRRLMMMMRb X6MLMMMRRLMRRLMMMRRb X6MLMMMRRLMRRLMMRRa X6MLMMMRRLMRRLMRRLa X6MLMMMRRLMRRLMRRLRb X6MLMMMRRLMRRLMRRRRb X6MLMMMRRLMRRLRLLMa X6MLMMMRRLMRRRRLMMa X6MLMMMRRLMRRRRLMMRb X6MLMMMRRLMRRRRLMRRb X6MLMMMRRLMRRRRLRRa X6MLMMMRRLMRRRRRRLa X6MLMMMRRLMRRRRRRLRb X6MLMMMRRLMRRRRRRRRb X6MLMMMRRLRLLLLLLMa X6MLMMMRRRRLMMMMMMa X6MLMMMRRRRLMMMMMMRb X6MLMMMRRRRLMMMMMRRb X6MLMMMRRRRLMMMMRRa X6MLMMMRRRRLMMMRRLa X6MLMMMRRRRLMMMRRLRb X6MLMMMRRRRLMMMRRRRb X6MLMMMRRRRLMMRLLMa X6MLMMMRRRRLMRRLMMa X6MLMMMRRRRLMRRLMMRb X6MLMMMRRRRLMRRLMRRb X6MLMMMRRRRLMRRLRRa X6MLMMMRRRRLMRRRRLa X6MLMMMRRRRLMRRRRLRb X6MLMMMRRRRLMRRRRRRb X6MLMMMRRRRLRLLLLMa X6MLMMMRRRRRRLMMMMa X6MLMMMRRRRRRLMMMMRb X6MLMMMRRRRRRLMMMRRb X6MLMMMRRRRRRLMMRRa X6MLMMMRRRRRRLMRRLa X6MLMMMRRRRRRLMRRLRb X6MLMMMRRRRRRLMRRRRb X6MLMMMRRRRRRLRLLMa X6MLMMMRRRRRRRRLMMa X6MLMMMRRRRRRRRLMMRb X6MLMMMRRRRRRRRLMRRb X6MLMMMRRRRRRRRLRRa X6MLMMMRRRRRRRRRRLa X6MLMMMRRRRRRRRRRLRb X6MLMMMRRRRRRRRRRRRb X6MLMMRLLLLLLLLLLMa X6MLMRRLMMMMMMMMMMa X6MLMRRLMMMMMMMMMMRb X6MLMRRLMMMMMMMMMRRb X6MLMRRLMMMMMMMMRRa X6MLMRRLMMMMMMMRRLa X6MLMRRLMMMMMMMRRLRb X6MLMRRLMMMMMMMRRRRb X6MLMRRLMMMMMMRLLMa X6MLMRRLMMMMMRRLMMa X6MLMRRLMMMMMRRLMMRb X6MLMRRLMMMMMRRLMRRb X6MLMRRLMMMMMRRLRRa X6MLMRRLMMMMMRRRRLa X6MLMRRLMMMMMRRRRLRb X6MLMRRLMMMMMRRRRRRb X6MLMRRLMMMMRLLLLMa X6MLMRRLMMMRRLMMMMa X6MLMRRLMMMRRLMMMMRb X6MLMRRLMMMRRLMMMRRb X6MLMRRLMMMRRLMMRRa X6MLMRRLMMMRRLMRRLa X6MLMRRLMMMRRLMRRLRb X6MLMRRLMMMRRLMRRRRb X6MLMRRLMMMRRLRLLMa X6MLMRRLMMMRRRRLMMa X6MLMRRLMMMRRRRLMMRb X6MLMRRLMMMRRRRLMRRb X6MLMRRLMMMRRRRLRRa X6MLMRRLMMMRRRRRRLa X6MLMRRLMMMRRRRRRLRb X6MLMRRLMMMRRRRRRRRb X6MLMRRLMMRLLLLLLMa X6MLMRRLMRRLMMMMMMa X6MLMRRLMRRLMMMMMMRb X6MLMRRLMRRLMMMMMRRb X6MLMRRLMRRLMMMMRRa X6MLMRRLMRRLMMMRRLa X6MLMRRLMRRLMMMRRLRb X6MLMRRLMRRLMMMRRRRb X6MLMRRLMRRLMMRLLMa X6MLMRRLMRRLMRRLMMa X6MLMRRLMRRLMRRLMMRb X6MLMRRLMRRLMRRLMRRb X6MLMRRLMRRLMRRLRRa X6MLMRRLMRRLMRRRRLa X6MLMRRLMRRLMRRRRLRb X6MLMRRLMRRLMRRRRRRb X6MLMRRLMRRLRLLLLMa X6MLMRRLMRRRRLMMMMa X6MLMRRLMRRRRLMMMMRb X6MLMRRLMRRRRLMMMRRb X6MLMRRLMRRRRLMMRRa X6MLMRRLMRRRRLMRRLa X6MLMRRLMRRRRLMRRLRb X6MLMRRLMRRRRLMRRRRb X6MLMRRLMRRRRLRLLMa X6MLMRRLMRRRRRRLMMa X6MLMRRLMRRRRRRLMMRb X6MLMRRLMRRRRRRLMRRb X6MLMRRLMRRRRRRLRRa X6MLMRRLMRRRRRRRRLa X6MLMRRLMRRRRRRRRLRb X6MLMRRLMRRRRRRRRRRb X6MLMRRLRLLLLLLLLMa X6MLMRRRRLMMMMMMMMa X6MLMRRRRLMMMMMMMMRb X6MLMRRRRLMMMMMMMRRb X6MLMRRRRLMMMMMMRRa X6MLMRRRRLMMMMMRRLa X6MLMRRRRLMMMMMRRLRb X6MLMRRRRLMMMMMRRRRb X6MLMRRRRLMMMMRLLMa X6MLMRRRRLMMMRRLMMa X6MLMRRRRLMMMRRLMMRb X6MLMRRRRLMMMRRLMRRb X6MLMRRRRLMMMRRLRRa X6MLMRRRRLMMMRRRRLa X6MLMRRRRLMMMRRRRLRb X6MLMRRRRLMMMRRRRRRb X6MLMRRRRLMMRLLLLMa X6MLMRRRRLMRRLMMMMa X6MLMRRRRLMRRLMMMMRb X6MLMRRRRLMRRLMMMRRb X6MLMRRRRLMRRLMMRRa X6MLMRRRRLMRRLMRRLa X6MLMRRRRLMRRLMRRLRb X6MLMRRRRLMRRLMRRRRb X6MLMRRRRLMRRLRLLMa X6MLMRRRRLMRRRRLMMa X6MLMRRRRLMRRRRLMMRb X6MLMRRRRLMRRRRLMRRb X6MLMRRRRLMRRRRLRRa X6MLMRRRRLMRRRRRRLa X6MLMRRRRLMRRRRRRLRb X6MLMRRRRLMRRRRRRRRb X6MLMRRRRLRLLLLLLMa X6MLMRRRRRRLMMMMMMa X6MLMRRRRRRLMMMMMMRb X6MLMRRRRRRLMMMMMRRb X6MLMRRRRRRLMMMMRRa X6MLMRRRRRRLMMMRRLa X6MLMRRRRRRLMMMRRLRb X6MLMRRRRRRLMMMRRRRb X6MLMRRRRRRLMMRLLMa X6MLMRRRRRRLMRRLMMa X6MLMRRRRRRLMRRLMMRb X6MLMRRRRRRLMRRLMRRb X6MLMRRRRRRLMRRLRRa X6MLMRRRRRRLMRRRRLa X6MLMRRRRRRLMRRRRLRb X6MLMRRRRRRLMRRRRRRb X6MLMRRRRRRLRLLLLMa X6MLMRRRRRRRRLMMMMa X6MLMRRRRRRRRLMMMMRb X6MLMRRRRRRRRLMMMRRb X6MLMRRRRRRRRLMMRRa X6MLMRRRRRRRRLMRRLa X6MLMRRRRRRRRLMRRLRb X6MLMRRRRRRRRLMRRRRb X6MLMRRRRRRRRLRLLMa X6MLMRRRRRRRRRRLMMa X6MLMRRRRRRRRRRLMMRb X6MLMRRRRRRRRRRLMRRb X6MLMRRRRRRRRRRLRRa X6MLMRRRRRRRRRRRRLa X6MLMRRRRRRRRRRRRLRb X6MLMRRRRRRRRRRRRRRb X6MLRLLLLLLLLLLLLMa X6MMMLMMMMMMMMMMMMa X6MMMLMMMMMMMMMMMMRb X6MMMLMMMMMMMMMMMRRb X6MMMLMMMMMMMMMMRRa X6MMMLMMMMMMMMMRRLa X6MMMLMMMMMMMMMRRLRb X6MMMLMMMMMMMMMRRRRb X6MMMLMMMMMMMMRLLMa X6MMMLMMMMMMMRRLMMa X6MMMLMMMMMMMRRLMMRb X6MMMLMMMMMMMRRLMRRb X6MMMLMMMMMMMRRLRRa X6MMMLMMMMMMMRRRRLa X6MMMLMMMMMMMRRRRLRb X6MMMLMMMMMMMRRRRRRb X6MMMLMMMMMMRLLLLMa X6MMMLMMMMMRRLMMMMa X6MMMLMMMMMRRLMMMMRb X6MMMLMMMMMRRLMMMRRb X6MMMLMMMMMRRLMMRRa X6MMMLMMMMMRRLMRRLa X6MMMLMMMMMRRLMRRLRb X6MMMLMMMMMRRLMRRRRb X6MMMLMMMMMRRLRLLMa X6MMMLMMMMMRRRRLMMa X6MMMLMMMMMRRRRLMMRb X6MMMLMMMMMRRRRLMRRb X6MMMLMMMMMRRRRLRRa X6MMMLMMMMMRRRRRRLa X6MMMLMMMMMRRRRRRLRb X6MMMLMMMMMRRRRRRRRb X6MMMLMMMMRLLLLLLMa X6MMMLMMMRRLMMMMMMa X6MMMLMMMRRLMMMMMMRb X6MMMLMMMRRLMMMMMRRb X6MMMLMMMRRLMMMMRRa X6MMMLMMMRRLMMMRRLa X6MMMLMMMRRLMMMRRLRb X6MMMLMMMRRLMMMRRRRb X6MMMLMMMRRLMMRLLMa X6MMMLMMMRRLMRRLMMa X6MMMLMMMRRLMRRLMMRb X6MMMLMMMRRLMRRLMRRb X6MMMLMMMRRLMRRLRRa X6MMMLMMMRRLMRRRRLa X6MMMLMMMRRLMRRRRLRb X6MMMLMMMRRLMRRRRRRb X6MMMLMMMRRLRLLLLMa X6MMMLMMMRRRRLMMMMa X6MMMLMMMRRRRLMMMMRb X6MMMLMMMRRRRLMMMRRb X6MMMLMMMRRRRLMMRRa X6MMMLMMMRRRRLMRRLa X6MMMLMMMRRRRLMRRLRb X6MMMLMMMRRRRLMRRRRb X6MMMLMMMRRRRLRLLMa X6MMMLMMMRRRRRRLMMa X6MMMLMMMRRRRRRLMMRb X6MMMLMMMRRRRRRLMRRb X6MMMLMMMRRRRRRLRRa X6MMMLMMMRRRRRRRRLa X6MMMLMMMRRRRRRRRLRb X6MMMLMMMRRRRRRRRRRb X6MMMLMMRLLLLLLLLMa X6MMMLMRRLMMMMMMMMa X6MMMLMRRLMMMMMMMMRb X6MMMLMRRLMMMMMMMRRb X6MMMLMRRLMMMMMMRRa X6MMMLMRRLMMMMMRRLa X6MMMLMRRLMMMMMRRLRb X6MMMLMRRLMMMMMRRRRb X6MMMLMRRLMMMMRLLMa X6MMMLMRRLMMMRRLMMa X6MMMLMRRLMMMRRLMMRb X6MMMLMRRLMMMRRLMRRb X6MMMLMRRLMMMRRLRRa X6MMMLMRRLMMMRRRRLa X6MMMLMRRLMMMRRRRLRb X6MMMLMRRLMMMRRRRRRb X6MMMLMRRLMMRLLLLMa X6MMMLMRRLMRRLMMMMa X6MMMLMRRLMRRLMMMMRb X6MMMLMRRLMRRLMMMRRb X6MMMLMRRLMRRLMMRRa X6MMMLMRRLMRRLMRRLa X6MMMLMRRLMRRLMRRLRb X6MMMLMRRLMRRLMRRRRb X6MMMLMRRLMRRLRLLMa X6MMMLMRRLMRRRRLMMa X6MMMLMRRLMRRRRLMMRb X6MMMLMRRLMRRRRLMRRb X6MMMLMRRLMRRRRLRRa X6MMMLMRRLMRRRRRRLa X6MMMLMRRLMRRRRRRLRb X6MMMLMRRLMRRRRRRRRb X6MMMLMRRLRLLLLLLMa X6MMMLMRRRRLMMMMMMa X6MMMLMRRRRLMMMMMMRb X6MMMLMRRRRLMMMMMRRb X6MMMLMRRRRLMMMMRRa X6MMMLMRRRRLMMMRRLa X6MMMLMRRRRLMMMRRLRb X6MMMLMRRRRLMMMRRRRb X6MMMLMRRRRLMMRLLMa X6MMMLMRRRRLMRRLMMa X6MMMLMRRRRLMRRLMMRb X6MMMLMRRRRLMRRLMRRb X6MMMLMRRRRLMRRLRRa X6MMMLMRRRRLMRRRRLa X6MMMLMRRRRLMRRRRLRb X6MMMLMRRRRLMRRRRRRb X6MMMLMRRRRLRLLLLMa X6MMMLMRRRRRRLMMMMa X6MMMLMRRRRRRLMMMMRb X6MMMLMRRRRRRLMMMRRb X6MMMLMRRRRRRLMMRRa X6MMMLMRRRRRRLMRRLa X6MMMLMRRRRRRLMRRLRb X6MMMLMRRRRRRLMRRRRb X6MMMLMRRRRRRLRLLMa X6MMMLMRRRRRRRRLMMa X6MMMLMRRRRRRRRLMMRb X6MMMLMRRRRRRRRLMRRb X6MMMLMRRRRRRRRLRRa X6MMMLMRRRRRRRRRRLa X6MMMLMRRRRRRRRRRLRb X6MMMLMRRRRRRRRRRRRb X6MMMLRLLLLLLLLLLMa X6MMMMMLMMMMMMMMMMa X6MMMMMLMMMMMMMMMMRb X6MMMMMLMMMMMMMMMRRb X6MMMMMLMMMMMMMMRRa X6MMMMMLMMMMMMMRRLa X6MMMMMLMMMMMMMRRLRb X6MMMMMLMMMMMMMRRRRb X6MMMMMLMMMMMMRLLMa X6MMMMMLMMMMMRRLMMa X6MMMMMLMMMMMRRLMMRb X6MMMMMLMMMMMRRLMRRb X6MMMMMLMMMMMRRLRRa X6MMMMMLMMMMMRRRRLa X6MMMMMLMMMMMRRRRLRb X6MMMMMLMMMMMRRRRRRb X6MMMMMLMMMMRLLLLMa X6MMMMMLMMMRRLMMMMa X6MMMMMLMMMRRLMMMMRb X6MMMMMLMMMRRLMMMRRb X6MMMMMLMMMRRLMMRRa X6MMMMMLMMMRRLMRRLa X6MMMMMLMMMRRLMRRLRb X6MMMMMLMMMRRLMRRRRb X6MMMMMLMMMRRLRLLMa X6MMMMMLMMMRRRRLMMa X6MMMMMLMMMRRRRLMMRb X6MMMMMLMMMRRRRLMRRb X6MMMMMLMMMRRRRLRRa X6MMMMMLMMMRRRRRRLa X6MMMMMLMMMRRRRRRLRb X6MMMMMLMMMRRRRRRRRb X6MMMMMLMMRLLLLLLMa X6MMMMMLMRRLMMMMMMa X6MMMMMLMRRLMMMMMMRb X6MMMMMLMRRLMMMMMRRb X6MMMMMLMRRLMMMMRRa X6MMMMMLMRRLMMMRRLa X6MMMMMLMRRLMMMRRLRb X6MMMMMLMRRLMMMRRRRb X6MMMMMLMRRLMMRLLMa X6MMMMMLMRRLMRRLMMa X6MMMMMLMRRLMRRLMMRb X6MMMMMLMRRLMRRLMRRb X6MMMMMLMRRLMRRLRRa X6MMMMMLMRRLMRRRRLa X6MMMMMLMRRLMRRRRLRb X6MMMMMLMRRLMRRRRRRb X6MMMMMLMRRLRLLLLMa X6MMMMMLMRRRRLMMMMa X6MMMMMLMRRRRLMMMMRb X6MMMMMLMRRRRLMMMRRb X6MMMMMLMRRRRLMMRRa X6MMMMMLMRRRRLMRRLa X6MMMMMLMRRRRLMRRLRb X6MMMMMLMRRRRLMRRRRb X6MMMMMLMRRRRLRLLMa X6MMMMMLMRRRRRRLMMa X6MMMMMLMRRRRRRLMMRb X6MMMMMLMRRRRRRLMRRb X6MMMMMLMRRRRRRLRRa X6MMMMMLMRRRRRRRRLa X6MMMMMLMRRRRRRRRLRb X6MMMMMLMRRRRRRRRRRb X6MMMMMLRLLLLLLLLMa X6MMMMMMMLMMMMMMMMa X6MMMMMMMLMMMMMMMMRb X6MMMMMMMLMMMMMMMRRb X6MMMMMMMLMMMMMMRRa X6MMMMMMMLMMMMMRRLa X6MMMMMMMLMMMMMRRLRb X6MMMMMMMLMMMMMRRRRb X6MMMMMMMLMMMMRLLMa X6MMMMMMMLMMMRRLMMa X6MMMMMMMLMMMRRLMMRb X6MMMMMMMLMMMRRLMRRb X6MMMMMMMLMMMRRLRRa X6MMMMMMMLMMMRRRRLa X6MMMMMMMLMMMRRRRLRb X6MMMMMMMLMMMRRRRRRb X6MMMMMMMLMMRLLLLMa X6MMMMMMMLMRRLMMMMa X6MMMMMMMLMRRLMMMMRb X6MMMMMMMLMRRLMMMRRb X6MMMMMMMLMRRLMMRRa X6MMMMMMMLMRRLMRRLa X6MMMMMMMLMRRLMRRLRb X6MMMMMMMLMRRLMRRRRb X6MMMMMMMLMRRLRLLMa X6MMMMMMMLMRRRRLMMa X6MMMMMMMLMRRRRLMMRb X6MMMMMMMLMRRRRLMRRb X6MMMMMMMLMRRRRLRRa X6MMMMMMMLMRRRRRRLa X6MMMMMMMLMRRRRRRLRb X6MMMMMMMLMRRRRRRRRb X6MMMMMMMLRLLLLLLMa X6MMMMMMMMMLMMMMMMa X6MMMMMMMMMLMMMMMMRb X6MMMMMMMMMLMMMMMRRb X6MMMMMMMMMLMMMMRRa X6MMMMMMMMMLMMMRRLa X6MMMMMMMMMLMMMRRLRb X6MMMMMMMMMLMMMRRRRb X6MMMMMMMMMLMMRLLMa X6MMMMMMMMMLMRRLMMa X6MMMMMMMMMLMRRLMMRb X6MMMMMMMMMLMRRLMRRb X6MMMMMMMMMLMRRLRRa X6MMMMMMMMMLMRRRRLa X6MMMMMMMMMLMRRRRLRb X6MMMMMMMMMLMRRRRRRb X6MMMMMMMMMLRLLLLMa X6MMMMMMMMMMMLMMMMa X6MMMMMMMMMMMLMMMMRb X6MMMMMMMMMMMLMMMRRb X6MMMMMMMMMMMLMMRRa X6MMMMMMMMMMMLMRRLa X6MMMMMMMMMMMLMRRLRb X6MMMMMMMMMMMLMRRRRb X6MMMMMMMMMMMLRLLMa X6MMMMMMMMMMMMMLMMa X6MMMMMMMMMMMMMLMMRb X6MMMMMMMMMMMMMLMRRb X6MMMMMMMMMMMMMLRRa X6MMMMMMMMMMMMMMMLa X6MMMMMMMMMMMMMMMLRb X6MMMMMMMMMMMMMMMMRa X6MMMMMMMMMMMMMMMRLa X X6 X0 X6R X6M X6ML X6MM X6MR X6MMR X6MLR X6MLM X6MMM X6MMMR X6MLMR X6MLRL X6MLMM X6MMML X6MMMM X6MMMMR X6MLMRR X6MLRLL X6MLMRRR X6MLMRRL X6MLRLLL X6MLMMR X6MLMMM X6MLMMRL X6MLMMMR X6MLMMMM X6MMMLM X6MMMLR X6MMMLMR X6MMMLRL X6MMMLMM X6MMMMM X6MMMMMR X6MMMMMMR X6MMMMMM X6MMMMML

edit4:
a pattern

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B1234567/S0a1hx2achx3abcdhx4achx5hx6a X

edit4:
again

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B1234567/S0o1o2mop X

also, can you please implement strobing rules???
edit5:
anther pattern

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B1234567/S0o1o2mop3gmop4gmop5mop6o7 X

[CARuler]

while thinking about CA in different neighborhoods, i realized that for any 2 state CA on tilling pattern {p,4} where p>4 all rules need a version of b2a to escape because generalized versions of b3i and b2c don't work to expand the bounding box.

also, can strobing rules be added to the current simulator? what about ruletables?
edit:
list of different strobing rulespaces:

• b0o/s - b0o123-h4567/s0a1234567
• b0a/s - b0a123456/s0123-x4567
• b0/s - b01234567/s0123456o
• b0o3h/s - b0o123456/s0a123-x4567
• b0o3h/s0o - b0o1234567/s0123456o
• b0a7/s - b0a123-h4567/s0a123-x4567
• b0a7/s3x - b0a1234567/s0123456o
• b0/s6a - b0123456/s0123-x456
• b0/s7 - b0123-h4567/s0a123456o7
• b0o3h7/s3x - b0o1234567/s0a123456o
• b0o3h/s0o6a - b0o123456/s0123-x456
• b0a3h7/s0o - b0a1234567/s0123-x456o
• b0a7/s3x7 - b0a123-h4567/s0a123456o7
• b03h/s7 - b0123456/s0a123-x456o7
• b07/s6a - b0123-h4567/s0a123-x456

edit2:
what about notation for oblique ships??

also, i have a challenge: create a ship that travels along great wall lines

[muzik]

B3/S23 on {7,3} just seems to settle into still lifes after a while:

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7$3$(a(B(a2(B(A2(b(A3(B|1(a(B|1))))))(A3(B|1(a2(B(a3(B|1))(a2(B|1))))(a(B(a2(B|1))(a3(B|1))(A3(B|1)))))(b(A3(B|1))(A2(b(A3(B|1))))))(a2(B|1))(a3(B|1))))(A2(b(A3(B(a2(B(a3(B|1(a2(B|1))))(a2(B(A3(B|1))))))))))(A3(b(A2(b(A2(b(A3(B|1(a(B|1))))))(A3(B(a2(B|1))))))))))(a2(B(a2(B(A3(b(A3(B|1))(A2(b(A3(B|1)))))(B(a(B(a2(B|1(a2(B|1))))))(a2(B(a3(B|1))))))(A2(b(A3(b(A2(b(A3(B|1)))))(B(a(B|1(a2(B|1))))))))))(A2(b(A2(b(A2(b(A3(B|1))))(A3(B|1))))))))(b(A2(b(A3(B(a2(B(a2(B|1(A3(B|1))))(a3(B|1))))))(A2(b(A3(B|1(a(B|1))(a2(B|1)))))))))(a3(B(a2(B(a2(B|1(a2(B|1))))))))(B(a2(B(A3(B|1(a(B(a3(B|1))(A3(B|1))(a2(B|1))))(a2(B|1)))(b(A3(B|1))(A2(b(A3(B|1))))))(A2(b(A3(B|1))))))(a3(B(a2(B|1))))(A2(b(A3(B|1))))(A3(B|1(a(B|1))(a2(B|1)))(b(A3(B|1)))))(A3(b(A3(B(a(B(a3(B(a2(B|1(a3(B|1))(a2(B|1))))))(a2(B(A2(b(A3(B|1))))))))))))

[CARuler]

playing with dots

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B1h2mo/S X

edit:

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B1h2mop/S X

edit2:

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B1h2o3o/S X

edit3:

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B1h2op3o/S X

edit4:

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B1h2o/S0o3o X

edit5:

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B1h2mo/S0o3o X

edit6:
osc

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B2ahox3m/S1ho X X1a X2

edit7:
seems like a good rule

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B2aho/S0o1ho2p3m

edit8:
osc

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B1h2a3bc/S2o5o X6a X5a X4a X3a X2a X1 X1Rb X

edit9:
osc

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B2aco/S0o2h X0a X1a X2

edit10:
ship

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B2ahox3dm/S1o2a X0a X1a X3a

edit11:
its p6 idk the offset or notation
edit12:
path

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B2ahox3dm/S1o2a X X3a X2Rb X2R X2RLa X2MRRb X2MRR X2MRRLa X2MRLRRb X2MRLRR X2MRLRRLa X2MRLRLRRb X2MRLRLRR X2MRLRLRRLa X2MRLRLRLRRb X2MRLRLRLRR X2MRLRLRLRRLa X2MRLRLRLRLRRb X2MRLRLRLRLRR

edit13:
p4 in that rule

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B2ahox3dm/S1o2a X6a X0a X1a X2a

edit14:
first ever gun!!!! (diff rule) (2 barreled)

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B1o2amx/S0a3o4x X X5a X3a

edit15:
p6 with that

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B1o2amx/S0a3ho4x X3Rb X1 X0R X1L X1La X4 X4Ma X4RL X4RLRa X4RLM X4RLR

someone pls process all collisions

[CARuler]

gun again (8 barreled)

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B1o2amox/S0a1x2c3bo4x X X5a X3a X0 X0La X0Ra

edit:
wait this replicates?! and its still a rake??!! what happens???

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B1o2amox/S0a1x2c3bo4x X0 X0Rb X X3a X3

edit2:
diff exponential growth (p7 idk the offset)

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B2aho/S1ho2a X X0a X1a

edit3:
max rule

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B2acho3abdghmx4cghmpx56a7/S1ho2a3bdghmp4567 X X0a X1a

edit4:
exponential puffer (p7 same offset)

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B2ao3d/S0o1ho2a X X0a X1a

help find the offset

[R2INT]

In hyperbolic space, it is possible for a moving replicator to replicate endlessly without interfering with itself.