ConwayLife.com

I was always wondering whether another „near-puffer-engine“ hides in such long running methuselahs and whether we would ever be able recognize it when it occurs. So something like a switch engine, only larger and easy to destroy because of a long cycle.

By now, billions of apgsearch runs would have ...

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 ...

Two other 1-loops

17x1, skew 4



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 ...

P9 (actually p3)



x = 24, y = 24, rule = B3/S23:T24,24
b2o4bo4b3o4b2o$
2b2o4b2o4bo4b3o$
2b3o4b2o4b2o4bo$
4bo4b3o4b2o4b2o$
o4b2o4bo4b3o4bo$
2o4b2o4b2o4bo4bo$
bo4b3o4b2o4b2o$
2b2o4bo4b3o4b2o$
3b2o4b2o4bo4b3o$
3b3o4b2o4b2o4bo$
o4bo4b3o4b2o4bo$
2o4b2o4bo4b3o$
3o4b2o4b2o4bo$
2bo4b3o4b2o4b2o ...

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



x = 20, y = 20, rule = B3/S23:T20,20
o3bo3bo3bo3bo$3bo3bo3bo3bo3bo$2bo3bo3bo3bo3bo$bo3bo3bo3bo3bo$
o3bo3bo3bo3bo$3bo3bo3bo3bo3bo$2bo3bo3bo3bo3bo$bo3bo3bo3bo3bo$
o3bo3bo3bo3bo ...

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 ...

Gemini says this pattern has just itself as predecessor on a 16x1 horizontal skew 7 torus



x = 16, y = 16, rule = B3/S23:T16,16
2o2bo3b2o2bo$b2o2bo3b2o2bo$2b2o2bo3b2o2bo$3b2o2bo3b2o2bo$o3b2o2bo3b2o$
bo3b2o2bo3b2o$2bo3b2o2bo3b2o$o2bo3b2o2bo3bo$2o2bo3b2o2bo$b2o2bo3b2o2bo$
2b2o2bo3b2o2bo ...

Another torus rle without predecessor von a 19x1 skew 3 torus. Quite likely it has predecessors on the 19x19.



x = 19, y = 19, rule = B3/S23:T19,19
3o2bobo2bob2obo$2bobo2bob2obo3b3o$bo2bob2obo3b3o2bo$bob2obo3b3o2bobo$2obo3b3o2bobo2bo$o3b3o2bobo2bob2o$b3o2bobo2bob2obo$o2bobo2bob2obo3b2o ...

The next step would be to extend such investigations on bigger tori, like 29x1, skew 12, which possesses a rotational symmetry.

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.

Not exactly high period, but nice:


x = 17, y = 1, rule = B3/S23:T17+4,1
11b6o!



x = 17, y = 17, rule = B3/S23:T17,17
11b6o$7b6o$3b6o$5o11bo$o11b5o$8b6o$4b6o$6o$2o11b4o$9b6o$5b6o$b6o$3o11b3o$10b6o$6b6o$2b6o$4o11b2o!


P7

x = 29, y = 1, rule = B3/S23:T29+12,1
13b4o8b4o!



x = 29, y = 29 ...

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 ...

#N Period-14 Resonator
#O Investigation of 1D Skewed Tori
#C Bounded to a 29x1 Torus with a horizontal shift s=+12
x = 29, y = 1, rule = B3/S23:T29+12,1
15b2o3bo2bo3b2o!





#N True Period-8 Breathing Oscillator (p=37, s=6)
#O Discovered via Python State-Space Explorer
#C Bounded to a 37x1 ...

Supposed "corrected version" based on your criticism. I skimmed it and it seems somewhat better.


Can this method be used to derive an even stronger bound for p2 and p3 oscillators? Can we use it to find out more on the space of possible density pairs resp. triples? Like limiting the maximum ...

I had a bunch of leftover credits, so I asked Claude Opus 5.5 to take a shot at the oscillating agar density conjecture; it did not solve it, but it claims to have established a new upper bound of 16/31 ≈ 0.516, much lower than amling's 1176/2087 bound ≈ 0.563.

Now, I have not verified this ...

If you were an AI, I would say “think harder”. If you go on a finite torus, then there are several ways to get factor 3. In the end this won’t give you something better for finite patterns in the infinite plane, but it’s not as obvious as you might think. This is the difference between a formal ...

This p32 is known?

Now that the case (m+n)/p < 1/2 is again fully solved, could we turn to the vastly different case of (m+n)/p=1/2. Here all known construction methods fail. Low period search engines also fail, even for (3,1)/8. The example of the L-wave shows that at least (2*n,2*m)/2*p can not have a simple local ...

While this is a very interesting patterns, it’s not p2. I changed direction and wanted to see what can actually be realized as density pairs.


Another one with density quotient 4/3



#N 15x5_Skew3_Gen0_Final
x = 15, y = 5, rule = B3/S23:T15,5
o5b3o4b2o$b3o5b3o$4b3o5b3o$3o4b3o$3b3o4b3o!





#N ...

The idea would be to in the end find the valid range of densities (d_0,d_1) for p2 oscillators.

A p2 with relatively high density, but different in different generations.

#N 17x1_Skew5_Gen1
#O AI_Collaborator
x = 17, y = 17, rule = B3/S23:T17,17
4b5o4b4o$4o4b4o4bo$3b4o4b5o$2o4b5o4b2o$b5o4b4o$o4b4o4b4o$4o4b5o$3b5o4b4o$
3o4b4o4b2o$2b4o4b5o$o4b5o4b3o$5o4b4o$4b4o4b5o$3o4b5o4bo$2b5o4b4o ...

You might not have recognized it, but I spent some time on the topic, see the patterns forum. In skew tori, there are basically three regimes that achieve maximum growth rates. The 3x3 torus with a block, the 3x1 with a stripe and 3x1 with skew 1. The last two can be mixed.

https://conwaylife.com ...

**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 ...

Gustone wrote: September 4th, 2026, 9:44 am

HartmutHolzwart wrote: September 4th, 2026, 2:44 amWe’re missing an engineered spaceship that is short and wide instead of long and thin

Shield bug, silk moth, geminoids...

Shield bug is not what I’d call “short”. I really wanted to say “has a horizontal spine although moving vertical”.

See the discussion and David Bell’s patterns