Patterns with unusual polynomial growth rates

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muzik
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Patterns with unusual polynomial growth rates

Post by muzik » May 5th, 2021, 1:52 am

By polynomial I mean growing at a rate t^x for any x above or equal to 0. There may be a better term for non-integer growth rates.

Basic examples:

t^0

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x = 1, y = 1, rule = B/S0
o!
[[ AUTOFIT GRAPH ]]
t^1

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x = 1, y = 1, rule = B1e/S01e2i3i4et
o!
[[ AUTOFIT GRAPH ]]
t^2

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x = 1, y = 1, rule = B1e3a/S03i4e5a6c7c8
o!
[[ AUTOFIT GRAPH ]]
And some slightly more exotic cases:

t^1/2
AforAmpere wrote:
April 27th, 2021, 3:44 am
Crazy sqrt growth:

Code: Select all

x = 3, y = 1, rule = B2-ck3c4ity5c6ci/S01e2e3nry4aeinqrt5-aekq6ace7e8
obo!
t^1/3
AforAmpere wrote:
April 28th, 2021, 12:21 pm
Appears to grow according to (EDIT, fixed) t^1/3:

Code: Select all

x = 3, y = 43, rule = B2aei3ekn4eikr5-aiqr6i8/S01e3nry4eijrwz5-ekr6c7e8
obo!

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yujh
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Re: Patterns with unusual polynomial growth rates

Post by yujh » May 5th, 2021, 2:14 am

t^1-t^(1/2)?

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x = 26, y = 47, rule = B34kz5e7c8/S23-a4ityz5k
18b2ob2o$16b2obobob2o$17b3ob3o$18bo3bo13$2b2o$bobo$bo$2o22$18bo3bo$17b
3ob3o$16bo2bobo2bo$16b2o5b2o$15b3o5b3o$15b3o5b3o!
Edit:
t^1??

Code: Select all

x = 3, y = 1, rule = B2-a3ajkq4aikqr5-kq6-n78/S01c2-i3ajn4-cikrt5aikn678
obo!
Rule modifier

B34kz5e7c8/S23-a4ityz5k
b2n3-q5y6cn7s23-k4c8
B3-kq6cn8/S2-i3-a4ciyz8
B3-kq4z5e7c8/S2-ci3-a4ciq5ek6eik7

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muzik
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Re: Patterns with unusual polynomial growth rates

Post by muzik » May 5th, 2021, 2:57 am

Some more examples:

t^1/3
AforAmpere wrote:
January 26th, 2021, 4:39 pm
This grows based on the cube root of t, with n dots past the o$bo$o! part by t=(2/3)n^3+2n^2+(16/3)n-3

Code: Select all

x = 6, y = 3, rule = B2ek3in4ce5ey6ci/S012ac3an4t5ei6c
o$bo$o!
t^1/2
wildmyron wrote:
January 13th, 2018, 11:54 am
Accidentally found this logarithmic growth whilst looking for spaceships.

Code: Select all

x = 4, y = 4, rule = B3aeij/S01c2n3ack4q5a6e
o$3bo$2bo$bobo!
BlinkerSpawn wrote:
January 13th, 2018, 4:26 pm
gameoflifemaniac wrote:
Majestas32 wrote:Isn't it square-root growth?
No.
This pattern adds 1 cell to the line with linearly increasing intervals. This means that we can think of the growth rate as the sum O(1+1/2+1/3+1/4+1/5...), which approximates logarithmic growth, not square-root growth. The O means that it grows asymptotically.
Let t be time and L be the length of the pattern.
The time between successive lengthenings is (roughly) 2L, so the growth rate of L is ~1/2L.
In calculus, this is written as dL/dt = 1/2L; a seperable differential equation.
Solving as usual, we have:
2L dL = dt
L^2 = t
L = t^0.5
It (and all similar signal-extending structure) exhibits square-root growth.
77topaz wrote:
February 4th, 2018, 4:25 pm
AforAmpere wrote:
danny wrote:What type of growth is this?:

Code: Select all

x = 57, y = 57, rule = B2e3-nr4r5cei6a78/S1c2ace3cjny4ay6e
3o$obo$3o52$54b3o$54bobo$54b3o!
It appears to be square root growth, based on a small sample regression.
Yeah, it looks like a sqrtgun, with the two halves gradually getting further apart and thus taking longer to interact with each other again.
77topaz wrote:
February 17th, 2018, 4:03 am
KittyTac wrote:Log growth in a related rule:

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x = 9, y = 3, rule = 1e2-en3-ci4ir5y7c/2c3-ciq4ak6ack7e/3
.7A$A3.BA2.A$.7A!
Nope, that's square-root growth.
Macbi wrote:
January 12th, 2019, 2:11 pm
Quantum Tunnel wrote:I don’t know what this pattern should be classified as. It travels on its own trail, creates a dot, and then travels backwards to create another dot. The population seems to increase slower and slower. I’m not sure if this has already been discovered.

Code: Select all

x = 3, y = 2, rule = B2-a5/S034
bo$obo!
The population of the pattern increases in a way that is proportional to the square root of the time. So these patterns are called sqrt-growth. This is a very nice example!
AforAmpere wrote:
April 11th, 2020, 11:35 pm
Square root growth from 3 cells:

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x = 3, y = 2, rule = B2a3ey4e5e6i/S01c2c3i4t
bo$obo!
toroidalet wrote:
December 10th, 2016, 9:04 pm
sqrt(t) growth:

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x = 16, y = 15, rule = B2ce3aiy/S12aei3r
9bo$8bo$9bo5$bo11bobo$obo11bo4$7bo$8bo$7bo!
83bismuth38 wrote:
July 8th, 2017, 12:25 pm

Code: Select all

x = 9, y = 9, rule = B2c3ae4ai56c/S2-kn3-enq4
3b3o$3bobo$4bo$2o2bo2b2o$ob2ob2obo$2o2bo2b2o$4bo$3bobo$3b3o!
A for awesome wrote:
July 30th, 2017, 10:11 am

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x = 7, y = 9, rule = B2c3aj4nrt5i6c78/S1c23enr4aet5-iq67
3bo$2bo2$bobo2bo$o2b3o$bobo2bo2$2bo$3bo!
AbhpzTa wrote:
February 23rd, 2018, 7:28 pm
SQRT(t) growth:

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x = 49, y = 26, rule = B3-cr4e6c7c/S23-aky4q5e6acn
42b3ob3o23$o$bo$3o!
LaundryPizza03 wrote:
July 27th, 2019, 1:44 am
Another one of those sqrt-growth patterns

Code: Select all

x = 3, y = 3, rule = B3aeij4q5q/S01c2aen3ajknr4aqw
b2o$2o$o!
Moosey wrote:
July 27th, 2019, 7:19 pm
1. Not explosive.
2. Supports the G
3. Preloaf is a 3-color sqrt-growth TM.

Code: Select all

x = 59, y = 26, rule = B3-ky4q5cq/S02aen3ajknr4aqw
b3o7b2ob3o5b2o$ob2o2b2o2b3obo2bob3obo$2o2b2ob5o2b4obo2bobo$5b3o2bo2b4o
2b2obo$bob4obo3b3ob2o2b2o2bo$o3b3o2bob3ob3o6bo$bo2b2ob2o2b3o3bob3obo$b
o2b4obo5b2o3b2o2bo$3bobo2b2obo2bo7b2o$ob2o4b3ob2obo5bo2bo$3o2bo3bo3b8o
b3o23bo8bo$ob2o2b3o4b2o2bob2o27b2o8bo$3bobob4o3b4obob4o24b2o5b3o$o2b2o
bob4o5bo2b2o$bo3bob2o4b2o4bo2bo$obo4b4o2bo4b2o3bo$3o3bob6o4bob3o$2b2ob
obo2b2o3bob2o2bob2o$bo2bo2b4o2b2ob4o3b2o$bob4o2b2o3bobob2o$b2o2b8o2b4o
b2o2bo$obobo3bob3ob2o7b2o$bo2bob2o2bo2bo3bo2bo$3o2bobo2b3ob2ob4obobo$o
2bo4bobob2o2bo2bo2b3o$5o4b2ob4ob3o2b2o!
Related to that other sqrtgrowth rule by laundrypizza03.
FWKnightship wrote:
September 12th, 2019, 8:29 am
Sqrt(t) growth:

Code: Select all

x = 3, y = 3, rule = 234i/2e3-nqr/3
.2A$2A$A!
Moosey wrote:
August 23rd, 2020, 8:13 am
SquishyBoi wrote:
August 23rd, 2020, 1:01 am
dani wrote:
August 22nd, 2020, 9:17 am
I'm not sure what to call this:

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x = 33, y = 7, rule = B2ce3aeikq4aeikw5c/S1367
o31bo$o31bo$bo29bo2$bo29bo$o31bo$o31bo!
technically some sort of breeder, maybe?
I wouldn't say that; its population grows at a rate of O(sqrt(n)) rather than O(n^2)

I'd call it a sqrtgrow.
AforAmpere wrote:
December 13th, 2020, 8:36 pm
EDIT 2, sqrt growth that is 2x faster in one direction than the other:

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x = 2, y = 3, rule = B3aeir4aet5eiy/S012cik3ey5ey6c
o$bo$o!
3x:

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x = 2, y = 3, rule = B3aeinr4aet5ei/S012ci3ey5ey
o$bo$o!
yujh wrote:
January 1st, 2021, 9:14 pm
Sqrt growth by silversmith:

Code: Select all

x = 16, y = 32, rule = S1e2-a3-cij4aijkrtz5cnqr/B2ce3aejnr4aj5y6a/3
bbbobooobobbbbbo$
bobbooooooobbooo$
oboooooboobobbbb$
boobbboobobbooob$
obbbbboboboobboo$
obobobbbbboboboo$
oobbobobbobobobb$
oboooobbbbbobbbb$
obbboobboobobooo$
obbbbobobobobbob$
ooobobboobbobbbb$
boboooobooooboob$
bbbobobobbboobbo$
bbobbooooooobbbb$
obboboobbbbobbob$
bbbbobobbboobbbo$
bbbbobobbboobbbo$
obboboobbbbobbob$
bbobbooooooobbbb$
bbbobobobbboobbo$
boboooobooooboob$
ooobobboobbobbbb$
obbbbobobobobbob$
obbboobboobobooo$
oboooobbbbbobbbb$
oobbobobbobobobb$
obobobbbbboboboo$
obbbbboboboobboo$
boobbboobobbooob$
oboooooboobobbbb$
bobbooooooobbooo$
bbbobooobobbbbbo!
AforAmpere wrote:
January 26th, 2021, 7:46 pm
Moving sqrt growth

Code: Select all

x = 2, y = 3, rule = B2ek3aijnr4ai5ey6c/S01c2ac3ei4aet5ei
o$bo$o!


t^3/2
AbhpzTa wrote:
February 23rd, 2018, 7:28 pm
t^1.5 growth:

Code: Select all

x = 25, y = 44, rule = B3-cr4e6c7c/S23-aky4q5e6acn
2bo$b2o$obo31$15b2o$14bobobo$14bo2bobo$14b2o2bo$16bo$17bo4bo$21bobo$
19bo2bo$20bo2b2o$21bo2bo$21b3o!

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wwei47
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Re: Patterns with unusual polynomial growth rates

Post by wwei47 » May 5th, 2021, 8:10 am

Um, a lot of those aren't polynomials. Aren't polynomials supposed to have nonnegative integer exponents? You might want to change the name.
Help me find high-period c/2 technology!
My guide: https://bit.ly/3uJtzu9
My c/2 tech collection: https://bit.ly/3qUJg0u
Overview of periods: https://bit.ly/3LwE0I5
Most wanted periods: 76,116

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muzik
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Re: Patterns with unusual polynomial growth rates

Post by muzik » May 5th, 2021, 8:27 am

wwei47 wrote:
May 5th, 2021, 8:10 am
Um, a lot of those aren't polynomials. Aren't polynomials supposed to have nonnegative integer exponents? You might want to change the name.
...which is exactly what I was asking in the original post?

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wwei47
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Re: Patterns with unusual polynomial growth rates

Post by wwei47 » May 5th, 2021, 8:30 am

muzik wrote:
May 5th, 2021, 8:27 am
...which is exactly what I was asking in the original post?
Sorry. I missed that.
Help me find high-period c/2 technology!
My guide: https://bit.ly/3uJtzu9
My c/2 tech collection: https://bit.ly/3qUJg0u
Overview of periods: https://bit.ly/3LwE0I5
Most wanted periods: 76,116

AforAmpere
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Re: Patterns with unusual polynomial growth rates

Post by AforAmpere » May 5th, 2021, 3:43 pm

Very unusual growth, but O(t^(1/3)):

Code: Select all

x = 2, y = 3, rule = B3aeiy4eir5ey/S012ce3j4ct6c
o$bo$o!
EDIT, more t^(1/3):

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x = 2, y = 3, rule = B2ek3aiy4ce5e6ci/S012-kn3in4acei6c
o$bo$o!

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x = 2, y = 3, rule = B2ek3iy4cet5y6i/S012-kn3aij4it5ei6i
o$bo$o!
This somehow approaches t^(1/2), and has a really weird pop-plot:

Code: Select all

x = 2, y = 3, rule = B2e3in4ce5e6i/S012-kn3ij4acei6c
o$bo$o!
EDIT 2, a t^(1/2) with a randomized return time due to internal XOR stuff:

Code: Select all

x = 2, y = 3, rule = B2e3in4ce5e6i/S012-kn3ij4ei6i
o$bo$o!
I manage the 5S project, which collects all known spaceship speeds in Isotropic Non-totalistic rules. I also wrote EPE, a tool for searching in the INT rulespace.

Things to work on:
- Find (7,1)c/8 and 9c/10 ships in non-B0 INT.
- EPE improvements.

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