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Re: Stable patterns which are not glider-constructible
Posted: September 5th, 2026, 7:26 am
by HartmutHolzwart
**Subject: Exploring the Preimage and Loop Landscape of Skew N x 1 Tori**
Hi everyone,
Following the landmark work on unconstructible still lifes and agar-based limitations by Ilkka Törmä and Ville Salo (
https://arxiv.org), I would like to propose a systematic look into a specific 1D projection of Conway's Game of Life: **collapsing periodic agars into an N × 1 Skew Torus.**
Many space-filling, periodic 2D agars naturally collapse into highly compact 1D structures via a coordinate lattice projection (essentially a structural Hadamard-type matrix reduction). By focusing specifically on these N × 1 Skew Tori with a horizontal skew factor s, the 8-cell Moore neighborhood maps onto a 1D ring of length N with the following coordinate offsets:
[-1, +1] U [s-1, s, s+1] U [-s-1, -s, -s+1] (mod N)
While this strips away the standard 2D visual plane, it provides a highly structured, lower-dimensional environment for analyzing global transition graphs. The system can be modeled using the group ring F_2[t]/(t^N - 1), and the state-transition graph collapses cleanly into a quotient graph under the Dihedral group D_N.
### Observations from Small Sizes
Brute-force checks on small universes show that varying N and the skew s radically alters the topology of the quotient graph:
* **N=17, s=3:** The state space is dominated by the All-Dead (0) basin (~96%). The remaining states collapse into pure, static 1D Still Lifes (Solitons).
* **N=17, s=4:** The All-Dead basin expands to over 97%, but the system unlocks a true **Period-16 cycle** in the quotient graph (Period-32 in raw space) that deforms and breathes through different cell counts (3 -> 5 -> 4 -> 9 ...).
* **N=14, s=3:** Even dimensions yield true 1-loop classes that are pure **Phoenix Agars** (cell overlap between consecutive generations is exactly zero).
### SAT Solving Advantages
Given that searching for specific preimage structures or long grandfather chains is generally PSPACE-hard, the linear 1D chain topology of these skew tori seems uniquely suited for SAT solvers.
Because the All-Dead basin is overwhelmingly dominant, random bit assignments trigger immediate conflicts during Unit Propagation. A CDCL solver should be able to prune the 2^N search space exceptionally fast. Additionally, symmetry-breaking clauses can be strictly derived from the D_N orbits.
### Questions for Discussion
I am interested in using this framework to study:
1. **Quantitative Entropies:** Can we find a formal bound for the exponential decay of non-Garden-of-Eden states as N scales?
2. **Isolated Cycles:** What is the distribution of true, strict d-loops (isolated cycles with an In-Degree of exactly 1)?
3. **Asymptotic Behavior:** What kind of hidden structural properties or unexpected attractor behaviors emerge as N grows significantly larger, particularly when N is prime?
I would love to get your thoughts on this, or collaborate on optimizing the CNF clause generation for these asymmetric 1D neighborhoods to test larger rings (N >= 31).
Cheers,
Hartmut
Re: Stable patterns which are not glider-constructible
Posted: September 10th, 2026, 3:08 pm
by glider_rider
Anivec wrote: August 25th, 2026, 6:14 pm
Are any of these infinite patches also self forcing? I assume they are all self forcing if my conjecture is true. These are all transformations of known patches, with a few exceptions.
EDIT:
All of these are self forcing. I did a quick check.
The pond agar is not self-forcing, it has the following predecessor:
Code: Select all
x = 6, y = 6, rule = B3/S23:T6,6
obo$o2b2o$o$3bobo$2obo$3bo!
I'm guessing what you meant is that it doesn't have any predecessors on a 3x3 torus.
Re: Stable patterns which are not glider-constructible
Posted: September 10th, 2026, 9:05 pm
by Anivec
glider_rider wrote: September 10th, 2026, 3:08 pm
Anivec wrote: August 25th, 2026, 6:14 pm
Are any of these infinite patches also self forcing? I assume they are all self forcing if my conjecture is true. These are all transformations of known patches, with a few exceptions.
EDIT:
All of these are self forcing. I did a quick check.
The pond agar is not self-forcing, it has the following predecessor:
Code: Select all
x = 6, y = 6, rule = B3/S23:T6,6
obo$o2b2o$o$3bobo$2obo$3bo!
Ah, I didn’t think it was truly self forcing.
Re: Stable patterns which are not glider-constructible
Posted: September 27th, 2026, 3:01 am
by 400spartans
153-cell unsynthesizable still life:
Code:
Select all
x = 22, y = 23, rule = B3/S23
13b2o$5bo6bobo$4bobo5bo6b2o$4bo2bo2b2obo2b2o2bo$2b2ob2obo2bo2bo2bobo$
2bo2bo2b2o2bob2o2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo3b2o$2bo2bo2b2o2b
ob2o2bo2bo$bobo2bo2bob2o2bob2obo$bob2ob2o2bo2bo2bo2bo$2bo2bo2b2o2bob2o
2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo$2bo2bo2b2o2bob2o2bo$bobo2bo2bob
2o2bob2o$o2b2ob2o2bo2bobo$obo2bo2b2o2bo2bo$bobo2bo2bob2obo$3b2ob2o2bo
2bo$2bo2bo5b2o$3bobo$4bo!
and a LifeHistory view of a 245-cell self-forcing patch for this still life:
Code:
Select all
x = 22, y = 23, rule = LifeHistory
13.2A$5.A6.A.A$4.A.A4.DCD5.2A$4.A2.C2DACDC2D2C2.A$2.2A.2CDC2DC2DC2DCD
A$2.A2.C2D2C2DCD2C2DC$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D.2A$2.A2DC
2D2C2DCD2C2DA2.A$.A.C2DC2DCD2C2DCD2CDA$.A.ACD2C2DC2DC2DC2DC$2.A.DC2D
2C2DCD2C2DCD$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D$2.A2DC2D2C2DCD2C2D
C$.A.C2DC2DCD2C2DC.2C$A.D2CD2C2DC2DC.A$A.C2DC2D2C2DC2.A$.ADC2DC2DCD2C
.A$3.ACD2C2DCD.A$2.A.DCD4.2A$3.A.A$4.A!
Re: Stable patterns which are not glider-constructible
Posted: September 27th, 2026, 3:28 am
by I6_I6
400spartans wrote: September 27th, 2026, 3:01 am
153-cell unsynthesizable still life:
Code:
Select all
x = 22, y = 23, rule = B3/S23
13b2o$5bo6bobo$4bobo5bo6b2o$4bo2bo2b2obo2b2o2bo$2b2ob2obo2bo2bo2bobo$
2bo2bo2b2o2bob2o2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo3b2o$2bo2bo2b2o2b
ob2o2bo2bo$bobo2bo2bob2o2bob2obo$bob2ob2o2bo2bo2bo2bo$2bo2bo2b2o2bob2o
2bo$4bo2bo2b2obo2b2o$3b2ob2o2bo2bo2bo$2bo2bo2b2o2bob2o2bo$bobo2bo2bob
2o2bob2o$o2b2ob2o2bo2bobo$obo2bo2b2o2bo2bo$bobo2bo2bob2obo$3b2ob2o2bo
2bo$2bo2bo5b2o$3bobo$4bo!
and a LifeHistory view of a 245-cell self-forcing patch for this still life:
Code:
Select all
x = 22, y = 23, rule = LifeHistory
13.2A$5.A6.A.A$4.A.A4.DCD5.2A$4.A2.C2DACDC2D2C2.A$2.2A.2CDC2DC2DC2DCD
A$2.A2.C2D2C2DCD2C2DC$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D.2A$2.A2DC
2D2C2DCD2C2DA2.A$.A.C2DC2DCD2C2DCD2CDA$.A.ACD2C2DC2DC2DC2DC$2.A.DC2D
2C2DCD2C2DCD$3.DC2DC2D2CDC2D2CD$3.2CD2C2DC2DC2DC2D$2.A2DC2D2C2DCD2C2D
C$.A.C2DC2DCD2C2DC.2C$A.D2CD2C2DC2DC.A$A.C2DC2D2C2DC2.A$.ADC2DC2DCD2C
.A$3.ACD2C2DCD.A$2.A.DCD4.2A$3.A.A$4.A!
Brilliant! That's a new record by 1 cell. How did you find that?
Re: Stable patterns which are not glider-constructible
Posted: September 27th, 2026, 10:50 am
by NNlk05
I6_I6 wrote: September 27th, 2026, 3:28 am
Brilliant! That's a new record by 1 cell. How did you find that?
They said on Discord:
music man (400spartans) wrote:
i decided to fire up the ol' unsynth life search program, no new population improvements, but found a few unsynthesizable still lifes that i thought were interesting:
...
153-cell unsynthesizable still life
...
Re: Stable patterns which are not glider-constructible
Posted: October 3rd, 2026, 5:03 am
by HartmutHolzwart
This might not fit very well well. I’m still investigating small nx1 skew tori and the evolution graph we get when restricting to equivalence classes under the torus symmetry group.
The 17x1, skew 4 torus has more symmetry than the usual dihedral, so the graph collapses to just 2072 equivalence classes. Exaclty on of them produces a 1-loop:
Code:
Select all
x = 17, y = 1, rule = B3/S23:T17+4,1
10b2obob2o!
Or mapped out on a 17x17 torus
Code:
Select all
x = 17, y = 17, rule = B3/S23:T17,17
10b2obob2o$6b2obob2o$2b2obob2o$bob2o10b2o$o10b2obobo$7b2obob2o$3b2obob2o$obob2o10bo$2o10b2obo$8b2obob2o$4b2obob2o$2obob2o$b2o10b2obo$9b2obob2o$5b2obob2o$b2obob2o$ob2o10b2o!
This is just an isolated pattern on the specific torus, but I guess it has only few predecessors on a full board and thus may lead to another still life that is non constructible. Could someone search for an actual predecessor?
Re: Stable patterns which are not glider-constructible
Posted: October 3rd, 2026, 6:16 am
by vilc
HartmutHolzwart wrote: October 3rd, 2026, 5:03 am
This is just an isolated pattern on the specific torus, but I guess it has only few predecessors on a full board and thus may lead to another still life that is non constructible. Could someone search for an actual predecessor?
Here is a predecessor, found with JLS :
Code:
Select all
x = 17, y = 17, rule = B3/S23:T17,17
10b2obob2o$6b2obob2o$2b2obob2o$bob2o10b2o$o9bobobobo$6bo5b2o$3b2o2b4o$
obobobobo7bo$2o7bobobobo$7b4o2b2o$4b2o5bo$2obobobo$b2o10b2obo$9b2obob
2o$5b2obob2o$b2obob2o$ob2o10b2o!
I found it by forcing most cells except for a small region in the center. Clearing larger regions seems to only produce variations on this predecessor (there are at least 15 of them), but there seems to be remarkably few of them considering the low density of this still-life. I am currently running more thorough searches.
Re: Stable patterns which are not glider-constructible
Posted: October 3rd, 2026, 6:50 am
by HartmutHolzwart
Thanks for that! I was musing whether there could exist patterns that have “few” predecessors (but not are not really self-forcing), and could still give rise to non-glider-constructible still-lifes. Very speculative.
Re: Stable patterns which are not glider-constructible
Posted: October 3rd, 2026, 9:56 am
by dvgrn
HartmutHolzwart wrote: October 3rd, 2026, 6:50 am
I was musing whether there could exist patterns that have “few” predecessors (but not are not really self-forcing), and could still give rise to non-glider-constructible still-lifes. Very speculative.
Yup, it seems possible that a still life can be found that doesn't contain any self-forcing patches, and yet all of its many non-still-life predecessors are grandfatherless, or great^N-grandfatherless for some N.
I'm thinking that this might be a more likely property as still lifes get larger -- like, patch together a large number of regions that each have only a few non-stable predecessors. But I really have no idea, of course. I don't think we collectively have very good intuition yet about what happens to predecessor counts for dense still lifes that are bigger than our current rewinding-search tools can conveniently handle.
Re: Stable patterns which are not glider-constructible
Posted: October 3rd, 2026, 4:14 pm
by Chris857
I also had a speculation that maybe some still life exists that somehow requires infinite gliders to create the sparks necessary to make it. Don't know if any such thing exists.
Re: Stable patterns which are not glider-constructible
Posted: October 4th, 2026, 10:24 am
by vilc
JLS just finished the search for all predecessors of this 17x17 board, and found 1140 of them. Sample predecessor, which less structure but noticeable motifs :
Code:
Select all
x = 17, y = 17, rule = B3/S23:T17,17
b2o4bo2bob3obo$3o6bo6bo$obob2ob2o3bo3bo$bo4b2o4bo2b2o$4bob2o$2bo2bobob
2ob3obo$o2b2obo4b2o$4bo4bob2o$3o7bobob2o$o4bob3obo4bo$o8bo4bobo$ob2ob
2o8bo$4b2o4bob3obo$2bob2o8bo$2bo2bob2ob2o$4bo4b2o3bo$ob2o3bob2o!
Edit:
Chris857 wrote: October 3rd, 2026, 4:14 pm
I also had a speculation that maybe some still life exists that somehow requires infinite gliders to create the sparks necessary to make it. Don't know if any such thing exists.
This would look like a glider-construction version of
American Dream, I guess. A first step would be to find a similar pattern with a still-life instead of a map of America. One would then attempt to synthesise the infinite trail of sparks in parallel, plus the space dust (assuming the infinite predecessor does not contain an orphan, which is unlikely if the space-dust region is large).
Re: Stable patterns which are not glider-constructible
Posted: October 5th, 2026, 2:26 am
by HartmutHolzwart
Another torus rle without predecessor von a 19x1 skew 3 torus. Quite likely it has predecessors on the 19x19.
Code:
Select all
x = 19, y = 19, rule = B3/S23:T19,19
3o2bobo2bob2obo$2bobo2bob2obo3b3o$bo2bob2obo3b3o2bo$bob2obo3b3o2bobo$2obo3b3o2bobo2bo$o3b3o2bobo2bob2o$b3o2bobo2bob2obo$o2bobo2bob2obo3b2o$obo2bob2obo3b3o$2bob2obo3b3o2bobo$b2obo3b3o2bobo2bo$bo3b3o2bobo2bob2o$2b3o2bobo2bob2obo$2o2bobo2bob2obo3bo$bobo2bob2obo3b3o$o2bob2obo3b3o2bo$ob2obo3b3o2bobo$obo3b3o2bobo2bobo$3b3o2bobo2bob2obo!
The other two are
Code:
Select all
x = 19, y = 19, rule = B3/S23:T19,19
2o2bobo2bobo2b2o$bobo2bobo2b2o3b2o$o2bobo2b2o3b2o2bo$obo2b2o3b2o2bobo$2b2o3b2o2bobo2bobo$o3b2o2bobo2bobo2bo$b2o2bobo2bobo2b2o$2bobo2bobo2b2o3b2o$bo2bobo2b2o3b2o2bo$bobo2b2o3b2o2bobo$o2b2o3b2o2bobo2bo$2o3b2o2bobo2bobo$2b2o2bobo2bobo2b2o$o2bobo2bobo2b2o3bo$obo2bobo2b2o3b2o$2bobo2b2o3b2o2bobo$bo2b2o3b2o2bobo2bo$b2o3b2o2bobo2bobo$3b2o2bobo2bobo2b2o!
And
Code:
Select all
x = 19, y = 19, rule = B3/S23:T19,19
obobo2bobobo2bobo$bo2bobobo2bobo2bobo$bobobo2bobo2bobobo$obo2bobo2bobobo2bo$2bobo2bobobo2bobobo$bo2bobobo2bobobo2bo$bobobo2bobobo2bobo$obo2bobobo2bobo2bo$2bobobo2bobo2bobobo$bobo2bobo2bobobo2bo$o2bobo2bobobo2bobo$obo2bobobo2bobobo$2bobobo2bobobo2bobo$bobo2bobobo2bobo2bo$o2bobobo2bobo2bobo$obobo2bobo2bobobo$bo2bobo2bobobo2bobo$bobo2bobobo2bobobo$o2bobobo2bobobo2bo!
Those look even more likely to have predecessors
Re: Stable patterns which are not glider-constructible
Posted: October 5th, 2026, 1:01 pm
by vilc
HartmutHolzwart wrote: October 5th, 2026, 2:26 am
Another torus rle without predecessor von a 19x1 skew 3 torus. Quite likely it has predecessors on the 19x19.
[...]
Those look even more likely to have predecessors
I looked for predecessors of these on a 19x19 grid, but this time I decided to be less lazy and switched to LLS (faster but has no native support for tori). I made a Golly script to generate search files, see at the end of the post. Assuming I made no mistake writing the script, the first and the third patterns are their only predecessor in the 19x19 torus, proved with ~1 minute searches each. The second pattern has 20 predecessors (with I guess 2 equivalence classes for translations and reflexions), found in ~10 minutes (see below, truncate to get 19x19 predecessors), still much less than the 17x17 pattern.
Code: Select all
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobobobobobboooboobb$
bbbbobobbbbboboobbbbbbbbb$
bbbbbbobobbbbbbboobobbbbb$
bbobobobooobooobbobobobbb$
bbobbbbbbbbbbbobobbobobbb$
bbbobbooobobbbobobobbbobb$
bbbbobbbobobobbbobooobbbb$
bbbbbobobbobobbbbbbbbbbbb$
bbbobobobobbooobooobobobb$
bbbbbbboboobbbbbbbobobbbb$
bbbobbbbbbboobobbbbbobobb$
bbbooobooobbobobobobobobb$
bbbbbbbbbobobbobobbbbbbbb$
bboobobbbbbobobbbobbooobb$
bbbobobobobobooobbobbbobb$
bbobbobobbbbbbbbbbbobobbb$
bbobobbbobbooobobobobobbb$
bbobooobbobbbobobbbbbobbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobobobobobboooboobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobobobobobbbobboobb$
bbbbobobbbbbobooobbobbbbb$
bbbobbobobbbbbbbbbbbobobb$
bbbobobbooobooobobobobobb$
bbboboobbbbbbbobobbbbbobb$
bbbbbbboobobbbbbobobbbbbb$
bbbooobbobobobobobooobobb$
bbbbbobobbobobbbbbbbbbbbb$
bbbbbobobobbbobbooobobbbb$
bbbobbbobooobbobbbobobobb$
bbbobbbbbbbbbbbobobbobobb$
bbbooobooobobobobobobbobb$
bbbbbbbbbobobbbbboboobbbb$
bboobobbbbbobobbbbbbboobb$
bbbobobobobobooobooobbobb$
bbobbobobbbbbbbbbbbobobbb$
bbobobbbobbooobobbbbbobbb$
bbobooobbobbbobobobobobbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobobobobobbbobboobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobbbbbobobbbobboobb$
bbbbobobobobobooobbobbbbb$
bbbobbobobbbbbbbbbbbobobb$
bbbobobbbobbooobobobobobb$
bbbobooobbobbbobobbbbbobb$
bbbbbbbbbbbobobbobobbbbbb$
bbbooobobobobobobbooobobb$
bbbbbobobbbbboboobbbbbbbb$
bbbbbbbobobbbbbbboobobbbb$
bbbobobobooobooobbobobobb$
bbbobbbbbbbbbbbobobbobobb$
bbbbobbooobobbbobobobbbbb$
bbobbobbbobobobbbobooobbb$
bbbbbbobobbobobbbbbbbbbbb$
bbobobobobobbooobooobobbb$
bbobbbbboboobbbbbbbobobbb$
bbobobbbbbbboobobbbbbobbb$
bbobooobooobbobobobobobbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobbbbbobobbbobboobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbobbooobobbbbbobobbbobbb$
bbbobbbobobobobobooobbobb$
bbbbobobbobobbbbbbbbbbbbb$
bbobobobobbbobbooobobobbb$
bbbbbbobooobbobbbobobbbbb$
bbobbbbbbbbbbbobobbobobbb$
bbooobooobobobobobobboobb$
bbbbbbbbobobbbbboboobbbbb$
bbobobbbbbobobbbbbbboobbb$
bbobobobobobooobooobbobbb$
bbbbobobbbbbbbbbbbobobbbb$
bbbobbbobbooobobbbobobobb$
bbbooobbobbbobobobbbobobb$
bbbbbbbbbobobbobobbbbbbbb$
bboobobobobobobbooobooobb$
bbbobobbbbboboobbbbbbbobb$
bbbbbobobbbbbbboobobbbbbb$
bbbobobooobooobbobobobobb$
bbbbbbbbbbbbbobobbobobbbb$
bbobbooobobbbbbobobbbobbb$
bbbobbbobobobobobooobbobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbobbbbbbbobobbbbboboobbb$
bbboobobbbbbobobbbbbbbobb$
bbbbobobobobobooobooobbbb$
bbbobbobobbbbbbbbbbbobobb$
bbbobobbbobbooobobbbobobb$
bbbobooobbobbbobobobbbobb$
bbbbbbbbbbbobobbobobbbbbb$
bbbooobobobobobobbooobobb$
bbbbbobobbbbboboobbbbbbbb$
bbbbbbbobobbbbbbboobobbbb$
bbbobobobooobooobbobobobb$
bbbobbbbbbbbbbbobobbobobb$
bbbbobbooobobbbbbobobbbbb$
bbobbobbbobobobobobooobbb$
bbbbbbobobbobobbbbbbbbbbb$
bbobobobobobbbobbooobobbb$
bbobbbbbobooobbobbbobobbb$
bbobobbbbbbbbbbbobobbobbb$
bbbbooobooobobobobobobbbb$
bbobbbbbbbobobbbbboboobbb$
bbboobobbbbbobobbbbbbbobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bboobooobbobobobobobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbobbbobobobbbobbooobobb$
bbbobobbbobooobbobbbobobb$
bbbobobbbbbbbbbbbobobbobb$
bbobbooobooobobobobobobbb$
bboobbbbbbbobobbbbboboobb$
bbbboobobbbbbobobbbbbbbbb$
bbobbobobobobobooobooobbb$
bbobobbobobbbbbbbbbbbobbb$
bbbbobobbbobbooobobbbbbbb$
bbobobooobbobbbobobobobbb$
bbbbbbbbbbbbobobbobobbbbb$
bbbbooobobobobobobbbobbbb$
bbobbbobobbbbbobooobbobbb$
bbbobobbobobbbbbbbbbbbobb$
bbbobobobbooobooobobobobb$
bbbbboboobbbbbbbobobbbbbb$
bbbbbbbbboobobbbbbobobbbb$
bboobooobbobobobobobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbobbooobobbbobobobbbobbb$
bbbobbbobobobbbobooobbobb$
bbbbobobbobobbbbbbbbbbbbb$
bbobobobobbooobooobobobbb$
bbbbbboboobbbbbbbobobbbbb$
bbobbbbbbboobobbbbbobobbb$
bbooobooobboboboboboboobb$
bbbbbbbbobobbobobbbbbbbbb$
bbobobbbbbobobbbobbooobbb$
bbobobobobobooobbobbbobbb$
bbbbobobbbbbbbbbbbobobbbb$
bbbobbbobbooobobobobobobb$
bbbooobbobbbobobbbbbobobb$
bbbbbbbbbobobbobobbbbbbbb$
bboobobobobobobbooobooobb$
bbbobobbbbboboobbbbbbbobb$
bbbbbobobbbbbbboobobbbbbb$
bbbobobooobooobbobobobobb$
bbbbbbbbbbbbbobobbobobbbb$
bbobbooobobbbobobobbbobbb$
bbbobbbobobobbbobooobbobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bboobooobbobobobobobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbobbbbbobobbbobbooobobb$
bbbobobobobooobbobbbobobb$
bbbobobbbbbbbbbbbobobbobb$
bbobbbobbooobobobobobobbb$
bbooobbobbbobobbbbboboobb$
bbbbbbbbobobbobobbbbbbbbb$
bbobobobobobobbooobooobbb$
bbobobbbbboboobbbbbbbobbb$
bbbbobobbbbbbboobobbbbbbb$
bbobobooobooobbobobobobbb$
bbbbbbbbbbbbobobbobobbbbb$
bbbbooobobbbobobobbbobbbb$
bbobbbobobobbbobooobbobbb$
bbbobobbobobbbbbbbbbbbobb$
bbbobobobbooobooobobobobb$
bbbbboboobbbbbbbobobbbbbb$
bbbbbbbbboobobbbbbobobbbb$
bboobooobbobobobobobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bboobbobbbobobobobobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbobobobobobbbobbooobobb$
bbbobbbbbobooobbobbbobobb$
bbbobobbbbbbbbbbbobobbobb$
bbobbooobooobobobobobobbb$
bboobbbbbbbobobbbbboboobb$
bbbboobobbbbbobobbbbbbbbb$
bbobbobobobobobooobooobbb$
bbobobbobobbbbbbbbbbbobbb$
bbobobobbbobbooobobbbobbb$
bbbbobooobbobbbobobobbbbb$
bbbbbbbbbbbbobobbobobbbbb$
bbobooobobobobobobbooobbb$
bbbbbbobobbbbboboobbbbbbb$
bbobbbbbobobbbbbbboobobbb$
bbobobobobooobooobbobobbb$
bbobobbbbbbbbbbbobobbobbb$
bbbbbobbooobobbbbbobobbbb$
bboobbobbbobobobobobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bboobbobbbobobbbbbobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbobobobobobbooobooobobb$
bbbobbbbboboobbbbbbbobobb$
bbbobobbbbbbboobobbbbbobb$
bbbobooobooobbobobobobobb$
bbbbbbbbbbbobobbobobbbbbb$
bbbooobobbbobobobbbobbobb$
bbbbbobobobbbobooobbobbbb$
bbobobbobobbbbbbbbbbbobbb$
bbobobobbooobooobobobobbb$
bbbboboobbbbbbbobobbbbbbb$
bbbbbbbboobobbbbbobobbbbb$
bbobooobbobobobobobooobbb$
bbbbbbobobbobobbbbbbbbbbb$
bbobbbbbobobbbobbooobobbb$
bbobobobobooobbobbbobobbb$
bbobobbbbbbbbbbbobobbobbb$
bbbbbobbooobobobobobobbbb$
bboobbobbbobobbbbbobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bboobooobobobobobobbooobb$
bbbbbbbobobbbbboboobbbbbb$
bbbobbbbbobobbbbbbboobobb$
bbbobobobobooobooobbobobb$
bbbobobbbbbbbbbbbobobbobb$
bbobbbobbooobobbbobobobbb$
bbooobbobbbobobobbboboobb$
bbbbbbbbobobbobobbbbbbbbb$
bbobobobobobobbooobooobbb$
bbobobbbbboboobbbbbbbobbb$
bbbbobobbbbbbboobobbbbbbb$
bbobobooobooobbobobobobbb$
bbbbbbbbbbbbobobbobobbbbb$
bbbbooobobbbbbobobbbobbbb$
bbobbbobobobobobooobbobbb$
bbbobobbobobbbbbbbbbbbobb$
bbbobobobbbobbooobobobobb$
bbbbbobooobbobbbobobbbbbb$
bbbbbbbbbbbbbobobbobobbbb$
bboobooobobobobobobbooobb$
bbbbbbbobobbbbboboobbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bboobbobbbobobobbbobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbobobobobobbooobooobobb$
bbbobbbbboboobbbbbbbobobb$
bbbobobbbbbbboobobbbbbobb$
bbbobooobooobbobobobobobb$
bbbbbbbbbbbobobbobobbbbbb$
bbbooobobbbbbobobbbobbobb$
bbbbbobobobobobooobbobbbb$
bbobobbobobbbbbbbbbbbobbb$
bbobobobbbobbooobobobobbb$
bbbbobooobbobbbobobbbbbbb$
bbbbbbbbbbbbobobbobobbbbb$
bbobooobobobobobobbooobbb$
bbbbbbobobbbbboboobbbbbbb$
bbobbbbbobobbbbbbboobobbb$
bbobobobobooobooobbobobbb$
bbobobbbbbbbbbbbobobbobbb$
bbbbbobbooobobbbobobobbbb$
bboobbobbbobobobbbobooobb$
bbbbbbbobobbobobbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobbbobobobbbobboobb$
bbbbobobobbbobooobbobbbbb$
bbbobbobobbbbbbbbbbbobobb$
bbbobobbooobooobobobobobb$
bbboboobbbbbbbobobbbbbobb$
bbbbbbboobobbbbbobobbbbbb$
bbbooobbobobobobobooobobb$
bbbbbobobbobobbbbbbbbbbbb$
bbbbbbbobobbbobbooobobbbb$
bbbobobobooobbobbbobobobb$
bbbobbbbbbbbbbbobobbobobb$
bbbbobbooobobobobobobbbbb$
bbobbobbbobobbbbbobooobbb$
bbbbbbobobbobobbbbbbbbbbb$
bbobobobobobbooobooobobbb$
bbobbbbboboobbbbbbbobobbb$
bbobobbbbbbboobobbbbbobbb$
bbobooobooobbobobobobobbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobbbobobobbbobboobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bboobooobobobobobobbooobb$
bbbbbbbobobbbbboboobbbbbb$
bbbobbbbbobobbbbbbboobobb$
bbbobobobobooobooobbobobb$
bbbobobbbbbbbbbbbobobbobb$
bbobbbobbooobobbbbbobobbb$
bbooobbobbboboboboboboobb$
bbbbbbbbobobbobobbbbbbbbb$
bbobobobobobobbbobbooobbb$
bbobobbbbbobooobbobbbobbb$
bbbbobobbbbbbbbbbbobobbbb$
bbbobbooobooobobobobobobb$
bbboobbbbbbbobobbbbbobobb$
bbbbboobobbbbbobobbbbbbbb$
bboobbobobobobobooobooobb$
bbbobobbobobbbbbbbbbbbobb$
bbbobobobbbobbooobobbbobb$
bbbbbobooobbobbbobobobbbb$
bbbbbbbbbbbbbobobbobobbbb$
bboobooobobobobobobbooobb$
bbbbbbbobobbbbboboobbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbobbooobobobobobobbbobbb$
bbbobbbobobbbbbobooobbobb$
bbbbobobbobobbbbbbbbbbbbb$
bbobobobobbooobooobobobbb$
bbbbbboboobbbbbbbobobbbbb$
bbobbbbbbboobobbbbbobobbb$
bbooobooobboboboboboboobb$
bbbbbbbbobobbobobbbbbbbbb$
bbobobbbobobobbbobbooobbb$
bbobobobbbobooobbobbbobbb$
bbbbobobbbbbbbbbbbobobbbb$
bbbobbooobooobobobobobobb$
bbboobbbbbbbobobbbbbobobb$
bbbbboobobbbbbobobbbbbbbb$
bboobbobobobobobooobooobb$
bbbobobbobobbbbbbbbbbbobb$
bbbbbobobbbobbooobobbbbbb$
bbbobobooobbobbbobobobobb$
bbbbbbbbbbbbbobobbobobbbb$
bbobbooobobobobobobbbobbb$
bbbobbbobobbbbbobooobbobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbboobobbbbbobobbbbbbb$
bbooobboboboboboboooboobb$
bbbbobobbobobbbbbbbbbbbbb$
bbbbobobobbbobbooobobbbbb$
bbobbbobooobbobbbobobobbb$
bbobbbbbbbbbbbobobbobobbb$
bbooobooobobobobobobboobb$
bbbbbbbbobobbbbboboobbbbb$
bbobobbbbbobobbbbbbboobbb$
bbobobobobobooobooobbobbb$
bbbbobobbbbbbbbbbbobobbbb$
bbbobbbobbooobobbbbbobobb$
bbbooobbobbbobobobobobobb$
bbbbbbbbbobobbobobbbbbbbb$
bboobobobobobobbbobbooobb$
bbbobobbbbbobooobbobbbobb$
bbobbobobbbbbbbbbbbobobbb$
bbobobbooobooobobobobobbb$
bboboobbbbbbbobobbbbbobbb$
bbbbbboobobbbbbobobbbbbbb$
bbooobboboboboboboooboobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbboobobbbbbobobbbbbbb$
bbooobboboboboboboooboobb$
bbbbobobbobobbbbbbbbbbbbb$
bbbbbbobobbbobbooobobbbbb$
bbobobobooobbobbbobobobbb$
bbobbbbbbbbbbbobobbobobbb$
bbbobbooobobobobobobbbobb$
bbbbobbbobobbbbbobooobbbb$
bbbbbobobbobobbbbbbbbbbbb$
bbbobobobobbooobooobobobb$
bbbbbbboboobbbbbbbobobbbb$
bbbobbbbbbboobobbbbbobobb$
bbbooobooobbobobobobobobb$
bbbbbbbbbobobbobobbbbbbbb$
bboobobbbobobobbbobbooobb$
bbbobobobbbobooobbobbbobb$
bbobbobobbbbbbbbbbbobobbb$
bbobobbooobooobobobobobbb$
bboboobbbbbbbobobbbbbobbb$
bbbbbboobobbbbbobobbbbbbb$
bbooobboboboboboboooboobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbobbbbbbbobobbbbboboobbb$
bbboobobbbbbobobbbbbbbobb$
bbbbobobobobobooobooobbbb$
bbbobbobobbbbbbbbbbbobobb$
bbbobobbbobbooobobbbbbobb$
bbbobooobbobbbobobobobobb$
bbbbbbbbbbbobobbobobbbbbb$
bbbooobobobobobobbbobbobb$
bbbbbobobbbbbobooobbobbbb$
bbobobbobobbbbbbbbbbbobbb$
bbobobobbooobooobobobobbb$
bbbboboobbbbbbbobobbbbbbb$
bbbbbbbboobobbbbbobobbbbb$
bbobooobbobobobobobooobbb$
bbbbbbobobbobobbbbbbbbbbb$
bbobbbobobobbbobbooobobbb$
bbobobbbobooobbobbbobobbb$
bbobobbbbbbbbbbbobobbobbb$
bbbbooobooobobobobobobbbb$
bbobbbbbbbobobbbbboboobbb$
bbboobobbbbbobobbbbbbbobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobobobobobboooboobb$
bbbbobobbbbboboobbbbbbbbb$
bbbbbbobobbbbbbboobobbbbb$
bbobobobooobooobbobobobbb$
bbobbbbbbbbbbbobobbobobbb$
bbbobbooobobbbbbobobbbobb$
bbbbobbbobobobobobooobbbb$
bbbbbobobbobobbbbbbbbbbbb$
bbbobobobobbbobbooobobobb$
bbbbbbbobooobbobbbobobbbb$
bbbobbbbbbbbbbbobobbobobb$
bbbooobooobobobobobobbobb$
bbbbbbbbbobobbbbboboobbbb$
bboobobbbbbobobbbbbbboobb$
bbbobobobobobooobooobbobb$
bbobbobobbbbbbbbbbbobobbb$
bbobobbbobbooobobbbobobbb$
bbobooobbobbbobobobbbobbb$
bbbbbbbbbbobobbobobbbbbbb$
bbooobobobobobobboooboobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
x = 25, y = 25, rule = B3/S23
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbobbboobbobobbobobboobbb$
bbboobbobobbobobboobbbobb$
bbbbobobbobobboobbboobbbb$
bbbobbobobboobbboobbobobb$
bbbobobboobbboobbobobbobb$
bbobboobbboobbobobbobobbb$
bboobbboobbobobbobobboobb$
bbbboobbobobbobobboobbbbb$
bbobbobobbobobboobbboobbb$
bbobobbobobboobbboobbobbb$
bbbbobobboobbboobbobobbbb$
bbbobboobbboobbobobbobobb$
bbboobbboobbobobbobobbobb$
bbbbboobbobobbobobboobbbb$
bboobbobobbobobboobbboobb$
bbbobobbobobboobbboobbobb$
bbobbobobboobbboobbobobbb$
bbobobboobbboobbobobbobbb$
bbbboobbboobbobobbobobbbb$
bbobbboobbobobbobobboobbb$
bbboobbobobbobobboobbbobb$
bbbbbbbbbbbbbbbbbbbbbbbbb$
bbbbbbbbbbbbbbbbbbbbbbbbb!
The Gollyt script I used (encodes the predecessor search on a torus in LLS format) :
Code: Select all
import golly as g
letters = "abcdefghijklmnopqrstuvwxyz"
def get_word(n): # base 26
if n == 0:
return "a"
w = ""
while n != 0:
w += letters[n % 26]
n //= 26
return w
def make_pred_gen(w, h): # first generation of pred pattern
"""
Row : letter(s)
Column : number
0 0 0 0 0 0 0
0 a0 a1 a2 a0 a1 0
0 b0 b1 b2 b0 b1 0
0 c0 c1 c2 c0 c1 0
0 a0 a1 a2 a0 a1 0
0 b0 b1 b2 b0 b1 0
0 0 0 0 0 0 0
"""
res = ("0 " * (w + 4)) + '\n'
for j in range(h + 2):
res += "0 "
for i in range(w + 2):
res += get_word(j % h) + str(i % w) + " "
res += "0\n"
res += ("0 " * (w + 4)) + '\n'
return res
def make_target_gen(w, h, bitmap):
"""
Example :
* * * * * * *
* * * * * * *
* * 0 1 0 * *
* * 1 0 1 * *
* * 1 1 1 * *
* * * * * * *
* * * * * * *
"""
res = (("* " * (w + 4)) + '\n') * 2
for j in range(h):
res += "* * "
for i in range(w):
res += str(bitmap[i][j]) + " "
res += "* *\n"
res += (("* " * (w + 4)) + '\n') * 2
return res
# Find dimensions of torus
rule = g.getrule()
if ":T" not in rule:
g.show("Error : rule should be on a torus.")
g.exit()
w, h = [int(x) for x in rule.split(':')[1][1:].split(',')]
# New origin coordinates (upper left corner)
ox = -(w//2)
oy = -(h//2)
cells = g.getcells(g.getrect())
cells = [(x - ox, y - oy) for (x, y) in zip(cells[::2], cells[1::2])]
bitmap = [[0] * h for _ in range(w)]
for x, y in cells:
bitmap[x][y] = 1
res = make_pred_gen(w, h) + '\n' + make_target_gen(w, h, bitmap)
g.setclipstr(res)
g.show("Search file pasted to clipboard.")
Just run it on a pattern inside a torus. It will generate an LLS input file and copy it to your clipboard. If you paste it in a file, say "in.txt", you can then run :
for finding two predecessors (may return one predecessor and then UNSAT if there are no other), or
for finding all predecessors.
Re: Stable patterns which are not glider-constructible
Posted: October 5th, 2026, 1:52 pm
by LuveelVoom
Hmm, has anyone looked into unsynthesizable spaceships yet?
Re: Stable patterns which are not glider-constructible
Posted: October 5th, 2026, 1:54 pm
by Anivec
LuveelVoom wrote: October 5th, 2026, 1:52 pm
Hmm, has anyone looked into unsynthesizable spaceships yet?
I have discussed the possibility of their existence but none are currently known.
Re: Stable patterns which are not glider-constructible
Posted: October 6th, 2026, 5:03 pm
by HartmutHolzwart
Gemini says this pattern has just itself as predecessor on a 16x1 horizontal skew 7 torus
Code:
Select all
x = 16, y = 16, rule = B3/S23:T16,16
2o2bo3b2o2bo$b2o2bo3b2o2bo$2b2o2bo3b2o2bo$3b2o2bo3b2o2bo$o3b2o2bo3b2o$
bo3b2o2bo3b2o$2bo3b2o2bo3b2o$o2bo3b2o2bo3bo$2o2bo3b2o2bo$b2o2bo3b2o2bo$
2b2o2bo3b2o2bo$3b2o2bo3b2o2bo$o3b2o2bo3b2o$bo3b2o2bo3b2o$2bo3b2o2bo3b2o$
o2bo3b2o2bo3bo!
Re: Stable patterns which are not glider-constructible
Posted: October 6th, 2026, 5:28 pm
by LuveelVoom
HartmutHolzwart wrote: October 6th, 2026, 5:03 pm
Gemini says this pattern has just itself as predecessor on a 16x1 horizontal skew 7 torus
What does LLS say, though?
Re: Stable patterns which are not glider-constructible
Posted: October 6th, 2026, 5:53 pm
by vilc
HartmutHolzwart wrote: October 6th, 2026, 5:03 pm
Gemini says this pattern has just itself as predecessor on a 16x1 horizontal skew 7 torus
This is true only up to a translation. On a 16x16 torus, the only predecessor of generation 0 of this pattern is generation 7 according to LLS. Have you tried running LLS? It is very helpful to answer this kind of question (it is not hard to implement predecessor searches in a skewed torus) without bothering to call a SAT-solver directly.
Also, I don't want to sound rude but since most of the wording of
this post of yours looks clearly AI-generated and you mention being assisted by AI in your research, could you please clarify the separation between your contributions and those of your LLM? Regarding that long post in particular, do you understand it all and can you explain, prove or provide reference for all the mathematical claims (the PSPACE-hardness of predecessor searches is not obvious, the advantage of modeling the space as F_2[t]/(t^N - 1) is not explicited, etc.). Also, what kind of algorithm did it use for finding the above pattern : brute-force exploration of the graph of evolutions in the skewed torus, randomised search, heuristic-based search or something else?
Re: Stable patterns which are not glider-constructible
Posted: October 7th, 2026, 2:23 am
by HartmutHolzwart
What I meant to say: This pattern is remarkable in that is actually a diagonally shifted copy of itself. On the 16x16 torus, it’s p8.
The pattern itself was generated by a Python script, which actually produced the full evolution graph of a 16x1 skew torus, where this pattern is an isolated component of. There are also 66 singular 1 loops in the graph that are actually still life agars, but I spared you these.
I’m actually trying to see whether on an nx1 skewed torus one can see certain CGOL aspects any clearer and actually also experimenting with what AI can actually do (or not). Of course, it is hallucinating a lot, but still sometimes, something interesting comes up.
Sorry for not being a native speaker… Most people are strangers in most cultures.
The post you refer to was AI generated. The idea was to somehow use algebraic geometry tools to get a different view on the problem, but it doesn’t seem to work. I worked on number fields in my youth, that’s why.
Edit:
Reading that old post again, I would phrase several parts differently. But one of the things I’m really interested in is understanding how many predecessors certain patterns have asymptotically with growing torus size given a fixed skew. Even understanding the all-dead basin is a challenge.
Re: Stable patterns which are not glider-constructible
Posted: October 8th, 2026, 7:25 am
by HartmutHolzwart
This one is an isolated 1-loop on the 20x1, skew 5 torus. Weird. I think I need to start SAT-solving on my own
Code:
Select all
x = 20, y = 20, rule = B3/S23:T20,20
o3bo3bo3bo3bo$3bo3bo3bo3bo3bo$2bo3bo3bo3bo3bo$bo3bo3bo3bo3bo$
o3bo3bo3bo3bo$3bo3bo3bo3bo3bo$2bo3bo3bo3bo3bo$bo3bo3bo3bo3bo$
o3bo3bo3bo3bo$3bo3bo3bo3bo3bo$2bo3bo3bo3bo3bo$bo3bo3bo3bo3bo$
o3bo3bo3bo3bo$3bo3bo3bo3bo3bo$2bo3bo3bo3bo3bo$bo3bo3bo3bo3bo$
o3bo3bo3bo3bo$3bo3bo3bo3bo3bo$2bo3bo3bo3bo3bo$bo3bo3bo3bo3bo!
Edit: 21x1, skew 8, p2 looks nice!
Code:
Select all
x = 21, y = 21, rule = B3/S23:T21,21
ob5ob2obobo$
2obobo7bob5o$
5bob5ob2obobo$
4ob2obobo7bobo$
obo7bob5ob2o$
2bob5ob2obobo$
ob2obobo7bob4o$
7bob5ob2obobo$
b5ob2obobo7bo$
obobo7bob5obo$
4bob5ob2obobo$
3ob2obobo7bob2o$
bo7bob5ob2obo$
bob5ob2obobo$
b2obobo7bob5o$
6bob5ob2obobo$
5ob2obobo7bo$
bobo7bob5ob2o$
3bob5ob2obobo$
2ob2obobo7bob3o$
o7bob5ob2obo!
24x1, skew 7 is above the Gemini limit
Re: Stable patterns which are not glider-constructible
Posted: October 8th, 2026, 12:30 pm
by HartmutHolzwart
Two other 1-loops
17x1, skew 4
Code:
Select all
x = 17, y = 17, rule = B3/S23:T17,17
o2b2o2bo4bo$
o2bo4bo4bo2bo$
4bo4bo2b2o2bo$
o4bo2b2o2bo$
bo2b2o2bo4bo$
2o2bo4bo4bo$
o4bo4bo2b2o$
bo4bo2b2o2bo$
2bo2b2o2bo4bo$
b2o2bo4bo4bo$
bo4bo4bo2b2o$
2bo4bo2b2o2bo$
3bo2b2o2bo4bo$
2b2o2bo4bo4bo$
2bo4bo4bo2b2o$
3bo4bo2b2o2bo$
4bo2b2o2bo4bo!
OK, I promise to leave it at that.
Re: Stable patterns which are not glider-constructible
Posted: October 9th, 2026, 12:32 am
by 400spartans
HartmutHolzwart wrote: October 6th, 2026, 5:03 pm
Gemini says this pattern has just itself as predecessor on a 16x1 horizontal skew 7 torus
This pattern is also self-forcing when tiled on an infinite plane.
I think there's plenty to find even in smaller toruses. For instance, the following "mango agar" pattern, which can also be represented as a 7x1 skew 2 or 5 torus, is self-forcing when tiled on an infinite plane.
Code:
Select all
x = 7, y = 7, rule = B3/S23:T7,7
2b2o2bo$bo2b2o$o2bo2bo$b2o2bo$o2b2o$2bo2b2o$2o2bo!
Re: Stable patterns which are not glider-constructible
Posted: October 9th, 2026, 2:34 am
by HartmutHolzwart
How hard is it to get from self-forcing still-life patterns on certain tori to self-forcing patterns on the plane? I‘m sort of naive on this point. I guess in most cases they will get non-self-forcing when going to the double torus, but I would think that if you enlarge the torus a few times and nothing new turns up, then your done. But‘s just a dumb intuition, not backed up by any sound theory.
Thanks a lot for taking this up!!! I‘m really curious how this works.
Re: Stable patterns which are not glider-constructible
Posted: October 9th, 2026, 10:07 pm
by 400spartans
The easiest thing to look for is an enlarged pattern which contains the original pattern in all of its preimages. This is the "self-enforcing" property mentioned by Törmä and Salo in their original paper. Check their discussion of the Köynnös agar to see how this property is shown and used in practice.