Infinite growth from 57o$73o!

[pcallahan]

The exclamation point is part of the RLE, but it is a little surprising (but based on an arbitrary encoding so not as mathematically interesting as other small pattern results). I was prepared to go into 3 digit runs before I found this:

Code: Select all

x = 73, y = 2, rule = B3/S23
57o$73o!

So this is 8 characters of RLE, including the terminator, that produces a stabilized switch engine (block making variety). Is this known? I tried searching on the RLE in quotes and did not find it.

It is a bit long to enter by hand, but doable. Pretty easy to remember 57 and 73 though. Are there shorter RLE strings that give infinite growth? It would require checking lines of cells up to 999999 in length. Has anyone tried it?

Same encoding length but infinite gliders.

Code: Select all

x = 73, y = 2, rule = B3/S23
57o$95o!

Both have a run of 57. I am not sure if there is a reason besides coincidence.

[dvgrn]

Are there shorter RLE strings that give infinite growth? It would require checking lines of cells up to 999999 in length. Has anyone tried it?

I haven't heard of it if so. Probably not, because the focus for infinite-growth patterns has generally been on minimizing population rather than RLE length.

I think you'd only have to check up to length 99999, though. "999999o" isn't any shorter than "57o$73o" ...

EDIT:

Same encoding length but infinite gliders... Both have a run of 57. I am not sure if there is a reason besides coincidence.

Definitely not a coincidence. That's the same exact switch engine in both cases, just a difference in the junk behind it throwing it into a different orbit.

[pcallahan]

I think you'd only have to check up to length 99999, though. "999999o" isn't any shorter than "57o$73o" ...

Oops, right. Hmm, going up to 99999 would not be too hard I think. At least there are not that many patterns to check.

[dvgrn]

I think you'd only have to check up to length 99999, though. "999999o" isn't any shorter than "57o$73o" ...

Oops, right. Hmm, going up to 99999 would not be too hard I think. At least there are not that many patterns to check.

I've exploded a fair number of straight width-1 lines in the past couple of decades, and have never seen a switch engine appear. Looks like for long lines a periodicity of 512 cells diagonally develops pretty quickly, superimposed on a 128-diagonal-cell periodicity, which is in turn superimposed on a 32-diagonal-cell periodicity.

So the main cases to worry about are the relatively few cases where the ends of the line are close enough to the tips of the explosion that unique interactions might happen. Those have probably all been tried many times already, but you never know. A universe that contains a 2-engine Cordership is capable of anything, really.

[pcallahan]

Both have a run of 57. I am not sure if there is a reason besides coincidence.

It's not a coincidence. The line of 57 cells followed by a longer line produces the switch engine in the same place. You can watch them together if you space them far enough apart (200 is not minimal, but it keeps them easy to separate by eye).

Code: Select all

x = 95, y = 154, rule = B3/S23
$57o$73o200$57o$95o!

Switch engines appear some time after generation 400. The context stabilizes the switch engines in different ways. It might be worth eyeballing the other bottom lines lengths to see why they fail.

[confocaloid]

An old thread, bumping for what may or may not be a sufficient reason. Here are both 57o$73o! and 57o$95o! from the above, overlapped in a "two layers" four-state CA where each of two layers follows the rules of Conway's Life:

Code: Select all

x = 95, y = 2, rule = DoubleB3S23
57C$73C22A!
#C [[ ZOOM 2 X -70 Y 50 ]]

Changing the length of the state-1 part of the bottom line lets to see how other lengths fail to make an infinite growth.

The RuleLoader .rule file (copied from Rule:DoubleB3S23), to make this more self-contained:

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@RULE DoubleB3S23

https://www.conwaylife.com/forums/viewtopic.php?p=22316#p22316

@TREE

num_states=4
num_neighbors=8
num_nodes=130
1 0 0 0 0
2 0 0 0 0
1 0 1 0 1
2 0 2 0 2
1 0 0 2 2
2 0 0 4 4
1 0 1 2 3
2 0 2 4 6
3 1 3 5 7
1 1 1 1 1
2 2 9 2 9
1 1 1 3 3
2 2 9 6 11
3 3 10 7 12
1 2 2 2 2
2 4 4 14 14
1 2 3 2 3
2 4 6 14 16
3 5 7 15 17
1 3 3 3 3
2 6 11 16 19
3 7 12 17 20
4 8 13 18 21
2 9 0 9 0
2 9 0 11 4
3 10 23 12 24
2 11 4 19 14
3 12 24 20 26
4 13 25 21 27
2 14 14 0 0
2 14 16 0 2
3 15 17 29 30
2 16 19 2 9
3 17 20 30 32
4 18 21 31 33
2 19 14 9 0
3 20 26 32 35
4 21 27 33 36
5 22 28 34 37
3 23 1 24 5
3 24 5 26 15
4 25 39 27 40
3 26 15 35 29
4 27 40 36 42
5 28 41 37 43
3 29 30 1 3
3 30 32 3 10
4 31 33 45 46
3 32 35 10 23
4 33 36 46 48
5 34 37 47 49
3 35 29 23 1
4 36 42 48 51
5 37 43 49 52
6 38 44 50 53
3 1 1 5 5
3 5 5 15 15
4 39 55 40 56
3 15 15 29 29
4 40 56 42 58
5 41 57 43 59
3 29 29 1 1
4 42 58 51 61
5 43 59 52 62
6 44 60 53 63
3 1 3 1 3
3 3 10 3 10
4 45 46 65 66
3 10 23 10 23
4 46 48 66 68
5 47 49 67 69
3 23 1 23 1
4 48 51 68 71
5 49 52 69 72
6 50 53 70 73
3 1 1 1 1
4 51 61 71 75
5 52 62 72 76
6 53 63 73 77
7 54 64 74 78
4 55 55 56 56
4 56 56 58 58
5 57 80 59 81
4 58 58 61 61
5 59 81 62 83
6 60 82 63 84
4 61 61 75 75
5 62 83 76 86
6 63 84 77 87
7 64 85 78 88
4 65 66 65 66
4 66 68 66 68
5 67 69 90 91
4 68 71 68 71
5 69 72 91 93
6 70 73 92 94
4 71 75 71 75
5 72 76 93 96
6 73 77 94 97
7 74 78 95 98
4 75 75 75 75
5 76 86 96 100
6 77 87 97 101
7 78 88 98 102
8 79 89 99 103
5 80 80 81 81
5 81 81 83 83
6 82 105 84 106
5 83 83 86 86
6 84 106 87 108
7 85 107 88 109
5 86 86 100 100
6 87 108 101 111
7 88 109 102 112
8 89 110 103 113
5 90 91 90 91
5 91 93 91 93
6 92 94 115 116
5 93 96 93 96
6 94 97 116 118
7 95 98 117 119
5 96 100 96 100
6 97 101 118 121
7 98 102 119 122
8 99 103 120 123
5 100 100 100 100
6 101 111 121 125
7 102 112 122 126
8 103 113 123 127
9 104 114 124 128

@TABLE

n_states:4
neighborhood:Moore
symmetries:permute

# 0x01 bit ON
var a={1,3}
var aa=a
var ab=a
var ac=a
var ad=a
var ae=a
var af=a
var ag=a

# 0x01 bit OFF
var b={0,2}
var ba=b
var bb=b
var bc=b
var bd=b
var be=b
var bf=b
var bg=b

# 0x10 bit ON
var c={2,3}
var ca=c
var cb=c
var cc=c
var cd=c
var ce=c
var cf=c
var cg=c

# 0x10 bit OFF
var d={0,1}
var da=d
var db=d
var dc=d
var dd=d
var de=d
var df=d
var dg=d

# don't-care cells
var x={0,1,2,3}
var xa=x
var xb=x
var xc=x
var xd=x
var xe=x
var xf=x
var xg=x
var xh=x

# birth or three-neighbor survival for both 0x01 and 0x10 bits
x,1,1,1,2,2,2,0,0,3
x,1,1,3,2,2,0,0,0,3
x,1,3,3,2,0,0,0,0,3
x,3,3,3,0,0,0,0,0,3

# survival for 0x01 bit, birth for 0x10 bit
1,2,2,2,1,1,0,0,0,3
1,2,2,3,1,0,0,0,0,3
1,2,3,3,0,0,0,0,0,3

# birth for 0x01 bit, survival for 0x10 bit
2,1,1,1,2,2,0,0,0,3
2,1,1,3,2,0,0,0,0,3
2,1,3,3,0,0,0,0,0,3

# 2-neighbor survival for 0x01 bit, 3-neighbor survival for 0x10 bit
3,1,1,2,2,2,0,0,0,3
3,1,3,2,2,0,0,0,0,3
3,3,3,2,0,0,0,0,0,3

# 3-neighbor survival for 0x01 bit, 2-neighbor survival for 0x10 bit
3,1,1,1,2,2,0,0,0,3
3,1,1,3,2,0,0,0,0,3
3,1,3,3,0,0,0,0,0,3

# 2-neighbor survival for both 0x01 and 0x10 bits
3,1,1,2,2,0,0,0,0,3
3,1,3,2,0,0,0,0,0,3
3,3,3,0,0,0,0,0,0,3

# birth for 0x01 bit, death for 0x10 bit
x,a,aa,ab,ba,bb,bc,bd,be,1
x,a,aa,ab,ba,bb,bc,bd,be,1
1,a,aa,ba,bb,bc,bd,be,bf,1
3,a,aa,ba,bb,bc,bd,be,bf,1

# death for 0x01 bit, birth for 0x10 bit
x,c,ca,cb,da,db,dc,dd,de,2
x,c,ca,cb,da,db,dc,dd,de,2
2,c,ca,da,db,dc,dd,de,df,2
3,c,ca,da,db,dc,dd,de,df,2

# death for both bits
x,xa,xb,xc,xd,xe,xf,xg,xh,0

@COLORS

0 0   0   0 
1 255 0   0
2 0   255 0
3 255 255 255