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Patterns with unusual growth rates

https://conwaylife.com/forums/viewtopic.php?t=5238

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Re: Patterns with unusual growth rates

Posted: October 18th, 2024, 2:08 pm
by CARuler
arbitrarily weird growth

Code: Select all

x = 9, y = 1, rule = growth
A.B.4FD!




@RULE growth
@TABLE
n_states:15
neighborhood:vonNeumann
symmetries:none
var a = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14}
var b = a
var c = a
var d = a
5,0,11,0,11,5
11,0,0,0,5,5

5,0,13,0,11,5
5,0,5,0,11,5
3,a,6,0,b,2
0,0,11,0,7,6
0,0,7,0,3,8
0,a,2,b,1,3
0,0,2,0,9,3
0,a,2,b,c,2
2,a,b,c,d,0
0,a,4,b,3,2
0,a,6,b,3,2
0,a,b,c,3,3
3,a,b,c,d,0
4,0,0,0,2,5
5,a,b,c,d,0
0,0,0,0,5,4
0,0,4,0,5,6
0,0,6,0,5,6
6,a,b,c,2,7
7,a,b,c,d,0
0,0,7,0,2,6
1,a,3,b,c,9
9,a,b,c,d,1
0,a,b,9,c,10
10,a,b,c,d,0
0,a,b,10,c,10
0,0,8,0,9,3
0,0,0,0,8,4
8,a,b,c,d,0
4,0,0,0,7,5
6,0,a,0,7,11
11,a,b,c,6,0
0,0,6,0,8,12
12,a,b,c,d,6
6,a,b,c,12,5
11,a,b,c,12,0
4,a,b,c,11,5
6,0,a,0,11,11
0,0,11,0,11,13
11,0,0,0,13,0
13,0,0,0,0,6
0,0,11,0,5,14
11,a,b,c,14,4
14,a,b,c,d,6
11,a,11,b,13,0
11,a,13,b,13,0
13,0,0,0,5,5

Re: Patterns with unusual growth rates

Posted: October 18th, 2024, 9:31 pm
by b-engine
CARuler wrote: October 18th, 2024, 2:08 pm arbitrarily weird growth

Code: Select all

x = 9, y = 1, rule = growth
A.B.4FD!
By utilizing the boundary and capture the graph, I found that it's actually just a linear growth:
screenshot (66).png
screenshot (66).png (31.94 KiB) Viewed 6593 times

Re: Patterns with unusual growth rates

Posted: October 18th, 2024, 10:25 pm
by CARuler
it may appear that way, but try running it golly for a very long time (sawtooth x times)

Re: Patterns with unusual growth rates

Posted: October 26th, 2024, 11:20 pm
by CARuler
is it possible to make a formula for this growth?

Code: Select all

x = 10, y = 10, rule = B2ae4e/S
7b2o$6bo2bo$5bo$4bo$3bo$2bo$bo5bo$o5bo$o$bo!
or this

Code: Select all

x = 10, y = 10, rule = B2ae4e/S
7b2o$6bo2bo$5bo$4bo$3bo$2bo5bo$bo5bo$o5bo$o4bo$bo!
or this

Code: Select all

x = 4, y = 4, rule = B2ae4e/S
b2o$o2bo$o$bo!
or this

Code: Select all

x = 11, y = 11, rule = B2ae4e/S
7b2o$6bo2bo$5bo$4bo5bo$3bo5bo$2bo5bo$bo5bo$o5bo$o4bo$bo2bo$3bo!

Re: Patterns with unusual growth rates

Posted: October 26th, 2024, 11:35 pm
by confocaloid
CARuler wrote: October 26th, 2024, 11:20 pm is it possible to make a formula for this growth? [...]
[...]

Code: Select all

x = 4, y = 4, rule = B2ae4e/S
b2o$o2bo$o$bo!
This one looks like there should be a reasonably simple expression for population = f(generation).
If you run it in Golly under HashLife algo with Hyperspeed enabled, it goes quickly in a 'regular' fashion without any messy ash or explosions.

The other ones look messy, and I'm not too optimistic about having a simple formula for one of them (but please do tell me otherwise).

Re: Patterns with unusual growth rates

Posted: October 27th, 2024, 8:05 pm
by vilc
confocaloid wrote: October 26th, 2024, 11:35 pm
CARuler wrote: October 26th, 2024, 11:20 pm is it possible to make a formula for this growth? [...]
[...]

Code: Select all

x = 4, y = 4, rule = B2ae4e/S
b2o$o2bo$o$bo!
This one looks like there should be a reasonably simple expression for population = f(generation).
The recursive structure is quite easy to translate into a recursive formula.
Let p(n) be the population at generation n. The behaviour of the sequence u(n) := p(n) - n - 6 is easier to understand since at generation n we have a frontend made of a diagonal row of n cells plus 6 cells on the edges.
From the evolution of the pattern, we have for n = 2^k + j, where 0 <= j < 2^k :
u(n) = 4*u(j) + j*(2^k - j)
A closed-form formula looks difficult to obtain in the general case, but here are expressions for p(n) at the beginning and end of each cycle of growth :
p(2^k) = 2^k + 6
p(2^k - 1) = (2^k - 1) + 6 + (2^k - 1)(2^(k-1) - 1)/3
Asymptotically :
lim inf p(n) / n = 1
lim sup p(n) / ((n^2)/6) = 1
Edit: Corrected asymptotics.
Edit 2: Corrected the formula for p(2^k - 1), which I had incorrectly written. See confocaloid's remark below.

Re: Patterns with unusual growth rates

Posted: October 27th, 2024, 8:08 pm
by CARuler
yes but at certain generations it doses not match this description

Re: Patterns with unusual growth rates

Posted: October 27th, 2024, 9:17 pm
by confocaloid
vilc wrote: October 27th, 2024, 8:05 pm [...]
Let p(n) be the population at generation n. The behaviour of the sequence u(n) := p(n) - n - 6 is easier to understand since at generation n we have a frontend made of a diagonal row of n cells plus 6 cells on the edges.
From the evolution of the pattern, we have for n = 2^k + j, where 0 <= j < 2^k :
u(n) = 4*u(j) + j*(2^k - j)
A closed-form formula looks difficult to obtain in the general case, but here are expressions for p(n) at the beginning and end of each cycle of growth :
p(2^k) = 2^k + 6
p(2^k - 1) = 2^k + 6 + (2^k - 1)(2^(k-1) - 1)
[...]
Values of the expressions for u(n) and p(2^k) agree with those derived from the population after directly evolving the pattern in Golly, but the expression for p(2^k - 1) looks incorrect. For example it gives 523783 for k = 10 (generation 1023), while the actual population is 175280.

Edit: I think the expression can be corrected as follows:
p(2^k - 1) = 2^k + 6 + (2^k - 1)(2^(k-1) - 1)/3 - 1
so the error may be due to incomplete copy&paste. This can be rewritten to
p(2^k - 1) = 6 + (2^k - 1)(2^k + 4) / 6

Re: Patterns with unusual growth rates

Posted: January 4th, 2025, 6:09 pm
by CARuler
lengthing ship gun(s)

Code: Select all

x = 1, y = 1, rule = lenghter
o!
@RULE lenghter
@COLORS
0   0   0   0
1 255 255 255
2 192 192 192
3 128 128 128
4  64  64  64
5 255   0   0
6   0 255   0
7   0   0 255
8  61  36  12
9 255 255   0
10 255   0 255
11   0 255 255
@TABLE
n_states:12
neighborhood:OneDimensional 
symmetries:none
var a = {0,1,2,3,4,5,6,7,8,9,10,11}
var b = a

0,1,0,5
0,6,0,7
0,7,0,5
8,4,7,9
8,10,6,11
8,10,8,11
0,6,10,7
8,0,8,0
0,7,11,5
9,5,8,0
5,8,9,0
8,0,6,9
8,10,0,0
8,8,0,0
8,0,0,0
0,7,9,5
0,6,11,7
#9,7,8,0

#defaults
1,a,b,2
2,a,b,3
3,a,b,4
4,a,b,1
5,a,b,6
6,a,b,8
7,a,b,8
9,a,b,10
10,a,b,0
11,a,b,9

Code: Select all

x = 1, y = 1, rule = lenghter
o!
@RULE lenghter
@COLORS
0   0   0   0
1 255 255 255
2 192 192 192
3 128 128 128
4  64  64  64
5 255   0   0
6   0 255   0
7   0   0 255
8  61  36  12
9 255 255   0
10 255   0 255
11   0 255 255
@TABLE
n_states:12
neighborhood:OneDimensional 
symmetries:none
var a = {0,1,2,3,4,5,6,7,8,9,10,11}
var b = a

0,1,0,5
0,6,0,7
0,7,0,5
8,4,7,9
8,10,6,11
8,10,8,11
0,6,10,7
8,0,8,0
0,7,11,5
9,5,8,0
5,8,9,0
8,0,6,9
8,10,0,0
8,8,0,0
8,0,0,0
0,7,9,5
0,6,11,7
9,7,8,0

#defaults
1,a,b,2
2,a,b,3
3,a,b,4
4,a,b,1
5,a,b,6
6,a,b,8
7,a,b,8
9,a,b,10
10,a,b,0
11,a,b,9
can anyone find the growth rate functions???????

Re: Patterns with unusual growth rates

Posted: January 14th, 2025, 10:33 pm
by eRroR_6o6
generally cursed growth i found in sticky

Code: Select all

x = 18, y = 13, rule = Sticky
.4A$A4.A$A3.BA6.A2.A$A3.CA5.A$A11.A2.A$.5A5.A$12.A4.A$.5A5.A$A11.A$A
3.CA5.A$A3.BA$A4.A$.4A!
it seems to grow at O(log(n)) after running it for 10^6 generations but I'm not sure

Re: Patterns with unusual growth rates

Posted: January 14th, 2025, 10:42 pm
by FWKnightship
This pattern I found in SoManyShips2 a while ago seems to be a O(log(n)) growth.

Code: Select all

x = 32, y = 27, rule = SoManyShips2
18.A$17.3A$17.BAB$8.2A$9.2A$6.B3.A$5.BA$7.AB$3.A3.B15.2A.B$3.2A18.2A$
4.2A20.B.B$23.B.B$27.2A$25.B.2A4$B.B.B.B$29.B$B.B.B.B$27.B3.B2$27.B3.
B2$27.B3.B2$29.B!

Re: Patterns with unusual growth rates

Posted: January 19th, 2025, 9:30 pm
by CARuler
puffes sawtooths

Code: Select all

x = 1, y = 3, rule = Sawthoother
A2$A!



@RULE Sawthoother
@TABLE
n_states:6
neighborhood:vonNeumann
symmetries:none
var a = {0,1,2,3,4,5}
var b = a
var c = a
var d = a

0,0,0,0,1,2
0,0,0,0,2,1
0,4,0,0,0,5
0,0,0,4,0,5
3,4,0,4,0,3
3,5,0,5,0,3
3,4,0,3,0,3
3,3,0,4,0,3
3,5,0,3,0,3
3,3,0,5,0,3
3,3,0,3,0,3
#defaults
1,a,b,c,d,4
2,a,b,c,d,0
3,a,b,c,d,0
4,a,b,c,d,3
5,a,b,c,d,4
edit:
a different type of sawtooth (being puffed)

Code: Select all

x = 1, y = 3, rule = SawWhat
o2$o!

@RULE SawWhat
@TABLE
n_states:6
neighborhood:vonNeumann
symmetries:none
var a = {0,1,2,3,4,5}
var b = a
var c = a
var d = a
0,0,0,0,1,2
0,0,0,0,2,1
0,4,0,0,0,5
0,0,0,4,0,5
3,4,0,3,0,5
3,3,0,4,0,5
3,3,0,3,0,3
3,4,0,4,0,3
3,5,0,5,0,3
3,4,0,3,0,3
3,3,0,4,0,3
3,5,0,3,0,3
3,3,0,5,0,3

#defaults
1,a,b,c,d,4
2,a,b,c,d,0
3,a,b,c,d,0
4,a,b,c,d,3
5,a,b,c,d,4
edit2:
muzik wrote: May 5th, 2021, 3:27 am [...]
EDIT 3, moves forward seemingly randomly, probably approaches some irrational constant like I suspect a lot of similar things do:

Code: Select all

x = 2, y = 3, rule = B2ek3in4cer6ci/S012ace3aei4aei5e6ci
o$bo$o!
[...]
no just pesdo-randomly
edit3:
finally using pesdo-randomness

Code: Select all

x = 1, y = 11, rule = SawWhat
A4$A2$A4$A!
@RULE SawWhat
@TABLE
n_states:6
neighborhood:vonNeumann
symmetries:none
var a = {0,1,2,3,4,5}
var b = a
var c = a
var d = a
0,0,0,0,1,2
0,0,0,0,2,1
0,4,0,0,0,5
0,0,0,4,0,5
3,4,0,3,0,5
3,3,0,4,0,5
3,3,0,3,0,3
3,4,0,4,0,3
3,5,0,5,0,3
3,4,0,3,0,3
3,3,0,4,0,3
3,5,0,3,0,3
3,3,0,5,0,3

#defaults
1,a,b,c,d,4
2,a,b,c,d,0
3,a,b,c,d,0
4,a,b,c,d,3
5,a,b,c,d,4

Re: Patterns with unusual growth rates

Posted: January 19th, 2025, 11:00 pm
by LuveelVoom
CARuler wrote: January 19th, 2025, 9:30 pm finally using pesdo-randomness

Code: Select all

x = 1, y = 11, rule = SawWhat
A4$A2$A4$A!
@RULE SawWhat
@TABLE
n_states:6
neighborhood:vonNeumann
symmetries:none
var a = {0,1,2,3,4,5}
var b = a
var c = a
var d = a
0,0,0,0,1,2
0,0,0,0,2,1
0,4,0,0,0,5
0,0,0,4,0,5
3,4,0,3,0,5
3,3,0,4,0,5
3,3,0,3,0,3
3,4,0,4,0,3
3,5,0,5,0,3
3,4,0,3,0,3
3,3,0,4,0,3
3,5,0,3,0,3
3,3,0,5,0,3

#defaults
1,a,b,c,d,4
2,a,b,c,d,0
3,a,b,c,d,0
4,a,b,c,d,3
5,a,b,c,d,4
I don’t think this is pseudo-random.
screenshot.png
screenshot.png (28.37 KiB) Viewed 6228 times

Re: Patterns with unusual growth rates

Posted: February 24th, 2025, 1:44 pm
by CARuler
ruler sequence

Code: Select all

x = 1, y = 1, rule = base2RulerSeq
o!
@RULE base2RulerSeq
@TABLE
n_states:6
neighborhood:vonNeumann
symmetries:none
var a = {0,1,2,3,4,5}
var b = a
var c = a
var d = a

0 0,0,1,0 2
0 0,0,2,0 1
0 0,0,0,1 3
0 0,0,3,2 3
0 0,0,3,1 5
0 0,0,0,5 3
0 0,0,3,0 3
0 0,0,3,3 3
0 0,0,3,5 5

#defaults
1,a,b,c,d,0
2,a,b,c,d,0
3,a,b,c,d,0
5,a,b,c,d,4
@COLORS
0   0   0   0
1 255 255   0
2 255   0 255
3   0 255 255
4 255 255 255
5 128  84  40

Re: Patterns with unusual growth rates

Posted: April 7th, 2025, 12:47 pm
by AforAmpere
Finally, t^1/4:

Code: Select all

x = 1, y = 4, rule = 012i3anry4r5ijq6ac7/2e3aijn4anrw5ry6ik/3
A$A$A$A!

Re: Patterns with unusual growth rates

Posted: April 7th, 2025, 1:48 pm
by confocaloid
AforAmpere wrote: April 7th, 2025, 12:47 pm Finally, t^1/4:

Code: Select all

x = 1, y = 4, rule = 012i3anry4r5ijq6ac7/2e3aijn4anrw5ry6ik/3
A$A$A$A!
The bounding box length sets records in generations
1, 5, 13, 23, 31, 47, 83, 105, 150, 178, 242, 276, 359, 399, 625, 671, 804, 856, 1393, 1451, 1646, 1710, 2720, 2790, 3059, 3135, 4816, 4898, 5253, 5341, 7927, 8021, 8474, 8574, 12335, 12441, 13004, 13116, 18358, 18476, 19161, 19285, 26350, 26480, 27299, 27435, 36701, 36843, 37808, 37956, 49837, 49991, 51114, 51274, 66220, 66386, 67679, 67851, 86348, 86526, 88001, 88185, 110755, 110945, 112614, 112810, 140011, 140213, 142088, 142296, 174722, 174936, 177029, 177249, 215530, 215756, 218079, 218311, 263113, 263351, 265916, 266160, 318185, 318435, 321254, 321510, 381496, 381758, 384843, 385111, 453832, 454106, 457469, 457749, 536015, 536301, 539954, 540246, 628903, 629201, 633156, 633460, 733390, 733700, 737969, 738285, 850406, 850728, 855323, 855651, 980917, 981251, 986184, 986524, 1125925, 1126271, 1131554, 1131906, 1286468, 1286826, 1292471, 1292835, 1463620, 1463990, 1470009, 1470385, 1658491, 1658873, 1665278, 1665666, 1872227, 1872621, 1879424, 1879824, 2106010, 2106416, 2113629, 2114041, ...
which appear to form four interleaving subsequences.

The following expressions (with integer n) should produce most of those subsequences, excluding the first few terms:

Code: Select all

(3*n^4 + 40*n^3 + 244*n^2 + 479*n + 484)/2
(3*n^4 + 40*n^3 + 244*n^2 + 503*n + 552)/2
(3*n^4 + 40*n^3 + 256*n^2 + 591*n + 718)/2
(3*n^4 + 40*n^3 + 256*n^2 + 615*n + 798)/2

Re: Patterns with unusual growth rates

Posted: June 9th, 2025, 10:00 pm
by CARuler
anther growth pattern

Code: Select all

x = 1, y = 1, rule = growth17
A!
@RULE growth17
@COLORS
0   0   0   0
1 255 255 255
2 255   0   0
3   0 255   0
4   0   0 255
5 255 255   0
6 255   0 255
7   0 255 255
8 128 128 128
9 255 128   0
10 128 255   0
11 255   0 128
12 128   0 255
13   0 255 128
14   0 128 255
@TABLE
n_states:15
neighborhood:vonNeumann
symmetries:none
var a = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14}
var b = a
var c = a
var d = a
var e = a
0,0,0,0,1,7
0,0,0,0,7,5
0,0,0,5,0,8
8,0,0,5,0,9
0,0,0,9,0,8
0,8,0,5,0,8
8,8,0,5,0,9
0,8,0,9,0,8
8,0,0,9,0,9
8,8,0,9,0,9
0,0,5,0,1,7
6,b,0,0,7,10
0,0,b,0,10,7
0,8,0,0,7,5

#defaults
1,b,c,d,e,2
2,b,c,d,e,3
3,b,c,d,e,4
4,b,c,d,e,1
5,b,c,d,e,6
6,b,c,d,e,5
8,b,c,d,e,8
a,b,c,d,e,0

Re: Patterns with unusual growth rates

Posted: June 23rd, 2025, 6:40 pm
by CARuler
FWKnightship wrote: January 14th, 2025, 10:42 pm This pattern I found in SoManyShips2 a while ago seems to be a O(log(n)) growth.

Code: Select all

x = 32, y = 27, rule = SoManyShips2
18.A$17.3A$17.BAB$8.2A$9.2A$6.B3.A$5.BA$7.AB$3.A3.B15.2A.B$3.2A18.2A$
4.2A20.B.B$23.B.B$27.2A$25.B.2A4$B.B.B.B$29.B$B.B.B.B$27.B3.B2$27.B3.
B2$27.B3.B2$29.B!
Viewing this at high speeds and zooming out reveals that this (seems to) be a binary counter which each decimal place being about O(2^n) cells from the starting point

Re: Patterns with unusual growth rates

Posted: October 19th, 2025, 3:49 pm
by CARuler
what is this counter?:

Code: Select all

x = 2, y = 3, rule = Countery
.B$AD$.B!


@RULE Countery
@COLORS
0   0   0   0
1 255 255 255
2   0   0 255
3   0 255   0
4 255   0   0
@TABLE
n_states:5
neighborhood:Moore
symmetries:rotate4reflect
var a = {0,1,2,3,4}
var b = a
var c = a
var d = a
var e = a
var f = a
var g = a
var h = a
var i = a

1,0,2,4,2,0,0,0,0,1
1,0,0,0,0,0,0,0,0,1
1,1,0,0,0,0,0,0,0,1
0,0,2,4,2,0,0,0,0,1
0,2,4,1,0,0,0,0,0,3
4,2,0,1,0,2,0,0,0,2
1,2,0,0,0,0,0,0,0,1
3,1,2,0,0,0,0,0,0,3
0,3,1,2,1,0,0,0,0,4
0,3,4,0,4,3,0,0,0,1
3,4,0,0,0,0,0,0,0,3
0,1,0,4,0,0,0,0,0,2
1,0,4,0,4,0,0,0,0,1
0,1,0,4,3,0,3,4,0,1
1,3,0,1,0,3,0,0,0,1
1,2,0,1,0,2,0,0,0,1
0,0,2,1,2,0,0,0,0,1
0,2,1,1,1,3,0,0,0,2
1,0,2,1,2,0,3,1,3,4
0,2,4,1,1,0,0,0,0,4
4,1,0,2,0,1,0,2,0,3
1,3,0,3,0,3,0,0,0,1
0,4,0,1,0,4,0,3,0,1
0,3,1,3,0,4,0,0,0,2
3,0,3,1,3,0,4,0,4,4
0,3,1,3,0,0,0,0,0,2
3,0,3,1,3,0,0,0,0,4
0,2,4,0,1,0,0,0,0,4
0,3,1,2,0,4,0,0,0,4
0,4,3,0,3,4,0,0,0,3
0,2,1,0,1,0,0,0,0,4
0,1,0,0,2,1,2,0,0,3
1,0,4,3,4,0,0,0,0,4
0,1,3,4,0,0,0,0,0,2
4,0,1,3,1,0,0,0,0,2
0,3,1,2,0,0,0,0,0,4
0,4,0,1,0,4,0,0,0,1

#defaults
a,b,c,d,e,f,g,h,i,0

Re: Patterns with unusual growth rates

Posted: November 2nd, 2025, 11:48 pm
by CARuler
seemingly random growth:

Code: Select all

x = 3, y = 5, rule = R2,C2,S5-6,B6,NW1000100500050500050010001
obo$obo2$obo$obo!

Code: Select all

x = 3, y = 2, rule = R2,C2,S5-6,B6,NW1000100500050500050010001
obo$obo!

Re: Patterns with unusual growth rates

Posted: November 6th, 2025, 8:38 pm
by CARuler
this should be here:
102564 wrote: November 6th, 2025, 7:21 pm What kind of growth is this? The population plot is wacky.

Code: Select all

x = 26, y = 5, rule = B3ai4ce5c6c7c/S3acein4et5i6cei78
2o$2o$11b2o11bo$11b3o9b3o$11b2o11bo!
p184

Code: Select all

x = 26, y = 26, rule = B3ai4ce5c6c7c/S3acein4eqt5i6cei78
2o20bo$2o19b3o$22bo10$22bo$21b3o$21b3o10$24b2o$24b2o!

Re: Patterns with unusual growth rates

Posted: March 16th, 2026, 12:27 am
by B468S02357
CARuler wrote: October 19th, 2025, 3:49 pm what is this counter?:

Code: Select all

x = 2, y = 3, rule = Countery
.B$AD$.B!


@RULE Countery
@COLORS
0   0   0   0
1 255 255 255
2   0   0 255
3   0 255   0
4 255   0   0
@TABLE
n_states:5
neighborhood:Moore
symmetries:rotate4reflect
var a = {0,1,2,3,4}
var b = a
var c = a
var d = a
var e = a
var f = a
var g = a
var h = a
var i = a

1,0,2,4,2,0,0,0,0,1
1,0,0,0,0,0,0,0,0,1
1,1,0,0,0,0,0,0,0,1
0,0,2,4,2,0,0,0,0,1
0,2,4,1,0,0,0,0,0,3
4,2,0,1,0,2,0,0,0,2
1,2,0,0,0,0,0,0,0,1
3,1,2,0,0,0,0,0,0,3
0,3,1,2,1,0,0,0,0,4
0,3,4,0,4,3,0,0,0,1
3,4,0,0,0,0,0,0,0,3
0,1,0,4,0,0,0,0,0,2
1,0,4,0,4,0,0,0,0,1
0,1,0,4,3,0,3,4,0,1
1,3,0,1,0,3,0,0,0,1
1,2,0,1,0,2,0,0,0,1
0,0,2,1,2,0,0,0,0,1
0,2,1,1,1,3,0,0,0,2
1,0,2,1,2,0,3,1,3,4
0,2,4,1,1,0,0,0,0,4
4,1,0,2,0,1,0,2,0,3
1,3,0,3,0,3,0,0,0,1
0,4,0,1,0,4,0,3,0,1
0,3,1,3,0,4,0,0,0,2
3,0,3,1,3,0,4,0,4,4
0,3,1,3,0,0,0,0,0,2
3,0,3,1,3,0,0,0,0,4
0,2,4,0,1,0,0,0,0,4
0,3,1,2,0,4,0,0,0,4
0,4,3,0,3,4,0,0,0,3
0,2,1,0,1,0,0,0,0,4
0,1,0,0,2,1,2,0,0,3
1,0,4,3,4,0,0,0,0,4
0,1,3,4,0,0,0,0,0,2
4,0,1,3,1,0,0,0,0,2
0,3,1,2,0,0,0,0,0,4
0,4,0,1,0,4,0,0,0,1

#defaults
a,b,c,d,e,f,g,h,i,0
Bine airy

Re: Patterns with unusual growth rates

Posted: April 1st, 2026, 8:50 pm
by Katrina
Woah

Code: Select all

x = 1, y = 1, rule = 11622909951
B!
@RULE 11622909951
@TABLE
n_states:3
neighborhood:oneDimensional
symmetries:none
var a = {0 1 2}
var b = a
var c = a
1,0,2,0
0,2,2,0
1,1,0,0
2,0,2,0
2,2,1,0
0,1,1,0
2,1,1,0
0,1,1,1
0,0,2,1
0,0,1,2
0,1,2,1
2,1,0,0
1,0,0,0
1,2,1,0
0,2,1,0
0,2,0,2
0,1,0,0
1,2,0,0
1,1,2,0
2,0,1,0
2,1,2,0
0,0,1,1
0,2,1,2
0,2,1,1
0,0,2,2
0,0,2,0
0,1,0,2
0,2,0,0
0,2,0,1
0,2,2,2
0,1,2,0
0,1,0,1
a,b,c,0

Re: Patterns with unusual growth rates

Posted: April 1st, 2026, 11:19 pm
by rabbit
This natural growth mechanism in a nearly-OT rule exhibits what appears to be noisy linear+sawtooth.

Code: Select all

x = 7, y = 9, rule = B2-a5678/S014567
4bo$2b2o$b3o$4o$4o$b4o$2b3obo$3b2o$3bobo!
Upon retraction the signal can fray into two different sections.

The rule is highly interesting in other aspects as well, and I may make a thread for it.

EDIT: It appears the frontend can be destroyed as well, meaning that the first pattern might eventually stop exhibiting growth and stabilize at an extremely high generation. Here is a variant which stabilizes at generation 586 for reference:

Code: Select all

x = 7, y = 9, rule = B2-a5678/S014567
4bo$2b2o$b3o$4o$2obo$b4o$2b3obo$3b2o$3bobo!
EDIT2: The first pattern breaks in somewhere around 150,000 generations.

EDIT3: I ran the first pattern to 2 million generations and it has not completely stabilized yet.

Re: Patterns with unusual growth rates

Posted: April 4th, 2026, 4:15 am
by yyh_baboon
n+sqrt(n) growth?

Code: Select all

 x = 3, y = 3, rule = B3aijr4j5e6i8/S2-ci3-aky4einrz5j6cn
3o$obo$2bo!
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