## Small Tori in B3/S23

For general discussion about Conway's Game of Life.
F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### Small Tori in B3/S23

Dear all,

I am interested in small tori in the 'normal' B3/S23 world. Most of what I've seen so far, in particular apgsearch/apgluxe, explores indefinite space starting with a 16x16 soup.

I'd like to get into contact with CGoL fans exploring (or having explored) tori like e.g. 8x8, or 16x16, or ...
I wonder whether an 8 x 8 B3/S23 torus can be or has been fully (exhaustively) explored (2^64 is a pretty large number for a brute-force approach, even if discounting all symmetries). If somebody can point me to software particularly suited for torus exploration (incl. oscillator detection), I would be very grateful. Or any link in this direction ... I have written simple C++ code for exploring tori up to 16x16, without 'ash' analysis (no dividing into individual objects), but with oscillator detection up to (theoretically) indefinitely large periods.

Disclaimer: I am not a mathematician, nor do I have any theoretical knowledge of cellular automata. My fascination with CGoL is probably mostly visual with a bit of 'treasure hunting' thrown in.

Hope to hear from you
Frank

x = 16, y = 9, rule = B3/S23:T16,16
o14bo$o14bo3$3b3o4b3o$5bo4bo$4bo6bo$7b2o$7b2o!

Rhombic
Posts: 1063
Joined: June 1st, 2013, 5:41 pm

### Re: Small Tori in B3/S23

Hello Frank and welcome to the community!
As far as I know, one of the earlier reports of ash (by Achim Flammenkamp) was run on 20x20 tori. This is not precisely small though, but in any case the proportion of objects reported almost certainly shows variation with respect to non-torus apgsearch.
As for the smaller 8x8 tori, I am not aware of such a brute-force attempt. I expect that with the known dynamics, unknown oscillators will be very unlikely but odds are there may well be many lurking in the agar shadows.

It would be interesting to see the evolutionary complexity of square tori (6x6 vs 5x5 vs 4x4 vs 3x3 and so on) because there may well be some general rules that haven't been explored, mostly due to the fact that most recent progress has been directed towards engineered patterns.

Surely an idea worth exploring.
SoL : FreeElectronics : DeadlyEnemies : 6a-ite : Rule X3VI
what is “sesame oil”?

Macbi
Posts: 714
Joined: March 29th, 2009, 4:58 am

### Re: Small Tori in B3/S23

We've exhaustively explored 6x6 starting configurations on an infinite grid, so doing all 6x6 tori should be possible. In fact I suspect 8x8 is also in reach because there are two factors that cut down on the search time. The first it that the large number of symmetries mean that fewer patterns need to be run, and the second is that each search stays contained in an 8x8 box rather than expanding. I suspect that you'll reach the limit some time before 16x16 will be though.

dvgrn
Moderator
Posts: 6302
Joined: May 17th, 2009, 11:00 pm
Contact:

### Re: Small Tori in B3/S23

Macbi wrote:We've exhaustively explored 6x6 starting configurations on an infinite grid, so doing all 6x6 tori should be possible. In fact I suspect 8x8 is also in reach because there are two factors that cut down on the search time... I suspect that you'll reach the limit some time before 16x16 will be though.
Yikes. I think the jump from 6x6 to 8x8 is an awfully big one. If 8x8 is possible, then someone should prove it by doing an exhaustive search on 7x7, which is 2^15 times easier than 8x8... and about eight thousand (2^13) times harder than 6x6, which is as big as we've gone so far.

Best not to even mention exhaustive searches on 16x16 tori -- at least, not until all the matter in the entire visible Universe has been converted to computronium. I forget whether that would be quite enough or not, I only remember that the topic came up recently, either here on the forums or on Discord.
F_rank wrote:I'd like to get into contact with CGoL fans exploring (or having explored) tori like e.g. 8x8, or 16x16, or ...

I wonder whether an 8 x 8 B3/S23 torus can be or has been fully (exhaustively) explored (2^64 is a pretty large number for a brute-force approach, even if discounting all symmetries). If somebody can point me to software particularly suited for torus exploration (incl. oscillator detection), I would be very grateful.
The existing software I know about is Gabriel Nivasch's RandomAgar, which can be found in the Life LInks thread. There's a predecessor, Jason Summers' torus.c, and a later Windows executable version called RandAgar, by Nicolay Beluchenko.

I don't think anyone has attempted, or even considered for very long, an exhaustive enumeration of all the oscillators that can appear from evolving 8x8 tori. The brute-force way of doing it involves quintillions of starting states, and Moore's Law or not, if you plan on getting a project done in your lifetime it had probably better not include the word "quintillions". Even "quadrillions", a thousand times easier, is pretty ambitious, whereas "trillions" is within reach of a good distributed search these days, if you have enough patience (or enough friends with computers).

Not sure how to get that 8x8 number of cases down to something searchable. Discarding symmetries won't do it. Skipping cases with, say, over 48 or fewer than 16 ON cells might be good as far as it goes (there may be interesting oscillators with phases in those extreme ranges, but it's pretty unlikely that all phases will be in those ranges).

But those turn out to be just insignificant trimmings at the far corners of the search space -- when you've thrown them all out, I think you're still stuck in the quintillions.

On the other hand, an exhaustive search on 7x7 tori is definitely somewhere in the doable range -- mere trillions of cases. That's my mantra: we don't need a Dyson sphere, we need a less ambitious threshold!

F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### Re: Small Tori in B3/S23

Wow, 3 qualified responses within one hour, I am overwhelmed! Thanks a lot.

No worries, I will start very small, learning as I go. Still have to learn a lot. Will keep you updated on my progress.

Frank

Macbi
Posts: 714
Joined: March 29th, 2009, 4:58 am

### Re: Small Tori in B3/S23

dvgrn wrote:
Macbi wrote:We've exhaustively explored 6x6 starting configurations on an infinite grid, so doing all 6x6 tori should be possible. In fact I suspect 8x8 is also in reach because there are two factors that cut down on the search time... I suspect that you'll reach the limit some time before 16x16 will be though.
Yikes. I think the jump from 6x6 to 8x8 is an awfully big one. If 8x8 is possible, then someone should prove it by doing an exhaustive search on 7x7, which is 2^15 times easier than 8x8... and about eight thousand (2^13) times harder than 6x6, which is as big as we've gone so far.
Yes, you are probably right. But 7x7 is definitely within reach.

Do you know what the average stabilisation time is for 6x6 soups on an unbounded grid is? I just tried some experiments on tori and for bounded grids (of side 6, 7, 8) the average stabilization time is around (11.4, 21.4, 31.0). If this is much smaller than for 6x6 unbounded-grid soups then that should help.

Also, since 7x7 and 8x8 tori contain less than 64 cells we can probably do some intricate bit-twiddling to compute each generation a bit faster.

That should get us up to 7x7. To get up to 8x8 I think we would have to do something cleverer than brute force, like perhaps keeping a hash table of some of the generations previously visited.

dvgrn
Moderator
Posts: 6302
Joined: May 17th, 2009, 11:00 pm
Contact:

### Re: Small Tori in B3/S23

Macbi wrote:Do you know what the average stabilisation time is for 6x6 soups on an unbounded grid is?
No clue here, except that it will be fairly fast. The vast majority of configurations will settle down to something p1 or p2 in just a few dozen ticks. So we're looking at somewhere between a few thousand soups per second at the low end, and probably a million soups per second on the high end.

It may be fast enough that it will be hard to think of ways to do complicated clever things, that will be any faster in practice than just running each soup to stability.
Macbi wrote:Also, since 7x7 and 8x8 tori contain less than 64 cells we can probably do some intricate bit-twiddling to compute each generation a bit faster.

That should get us up to 7x7. To get up to 8x8 I think we would have to do something cleverer than brute force, like perhaps keeping a hash table of some of the generations previously visited.
The problem with mixing hash tables with quintillions of cases is that you're likely to need a ridiculous amount of RAM. You might not start seeing a lot of speedup until you've stored trillions of hashes.

If so (I'm not saying it _is_ so, just making things up as I go along) then you might need a petabyte of RAM for your hash table.

Maybe a different way to think about it is this: hash tables aren't going to do you any good if you still have to iterate through anywhere near 2^64 cases. If you do any processing at all on each case, then that's already too much. Even checking symmetries to see if a given 8x8 is canonical or if you can skip it, will probably take considerably more than... let's say... a nanosecond (a billionth of a second).

Even at one nanosecond per 8x8 torus soup, we're looking at something over 30,000 CPU-years to get through all cases. (Right?)

Maybe there's a way to lump huge chunks of the search space together and discard them simultaneously. But given how unpredictable Life is, I'm not too optimistic. Generally the fastest way we know to find out what will happen to a pattern is to simulate it and see.

The big case where HashLife trickery can't improve on some simple fast bit-twiddling algorithm like QuickLife, is when you're dealing with pure randomness. And if you're looking through all possible cases, pure randomness is (on average) exactly what you're dealing with.

calcyman
Posts: 2129
Joined: June 1st, 2009, 4:32 pm

### Re: Small Tori in B3/S23

These are square tori, so you can modulo out by translations and rotations -- in the 7x7 case, that means you only have about (2 ** 49) / (8 * 49) = 1.43 trillion soups, which is about 7 times less than the number of soups searched in Catagolue. And these small finite soups are much faster to search than 16x16 seeds in an infinite universe.

In the 8x8 case, you have just over 2 ** 55 soups, and I fall into Dave's pessimistic camp with regards to that -- it's possible, but might just require four acres of Crays to accomplish in a sensible amount of time.
What do you do with ill crystallographers? Take them to the mono-clinic!

dvgrn
Moderator
Posts: 6302
Joined: May 17th, 2009, 11:00 pm
Contact:

### Re: Small Tori in B3/S23

calcyman wrote:These are square tori, so you can modulo out by translations and rotations...
And reflections?
calcyman wrote:In the 8x8 case, you have just over 2 ** 55 soups, and I fall into Dave's pessimistic camp with regards to that -- it's possible, but might just require four acres of Crays to accomplish in a sensible amount of time.
It's good to see that we're getting some consistency in our wild estimates. Given that a 16^2 torus would be a gazillion bazillion times harder to exhaustively search than an 8x8 one -- about 2^200 times harder, let's say, counting the 192 extra cells to be permuted, plus the factor of four increase in RAM needed to store each torus configuration -- notice that

"four acres of Crays"

times

"a gazillion bazillion"

equals exactly

"the entire visible universe converted to computronium".

What a coincidence!

F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### Re: Small Tori in B3/S23

Great discussion, tears of laughter running down my face

Adding to the irony is the fact (true!) that my 16x16 torus work started on an Arduino Uno with 4 'PixBlocks', each 8x8 dual LEDs (green/red which also allows for yellow/orange). This thing does maybe 10 (read: ten) soups per second ... what a perfect fit to the previous post ... Sometimes it is a small bridge from an Arduino Uno to a few acres of Crays.

-F

F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### Re: Small Tori in B3/S23

Guys, I may need your help with the established terminology, I am still a newbie in your world. Please correct me if you think a different term should be used.

So I looked into the "canonical" states (states? hmm...), in other words: by how much can we reduce our initial 2^(torus size) 'soups' (they are not soups anymore if we go brute force, are they?) when we factor in all mirroring, turning, shifting (important for tori!), ...

So this is my initial table (I will add more fields to each torus as I go along ...):
• Code: Select all

torus size     #cells         #states  #canonical
1           1               2           2
2           4              16           6
3           9             512          26
4          16          65,536         805
5          25      33,554,432     172,112
6          36  68,719,476,736         ???

The #canonical numbers are based on my own C++ code, which collapsed at torus size = 6 = 2^36 states. I used a self-conceived algorithm somewhat similar to the Sieve of Eratosthenes.

My first objective will be the analysis of the three tori of size 3, 4, and 5 (didn't I tell you I would start small?)

Currently, I am considering to describe each torus with the following characteristics, in addition to the ones listed above (absolute and relative [%] where applicable):
• # of empty final states
# of stable final states (period = 1)
# number of 'canonical' stable final states (potentially)
# of oscillating states / 'canonical' oscillating states
max. period (of all oscillating patterns observed)
average # of generations until stable or oscillating (including one full oscillating cycle)
max. # of generations until stable or oscillating
Any other ideas? Feedback most welcome.

-F

calcyman
Posts: 2129
Joined: June 1st, 2009, 4:32 pm

### Re: Small Tori in B3/S23

You can get a closed form for the number of canonical tori by either using Burnside's Lemma, or by typing your largest experimental value 172112 into OEIS and clicking on the first result:

http://oeis.org/A255016
What do you do with ill crystallographers? Take them to the mono-clinic!

Macbi
Posts: 714
Joined: March 29th, 2009, 4:58 am

### Re: Small Tori in B3/S23

I don't know how much effort this would be, but one thing I'd like to know is the entropy.

Start with a uniform probability over the possible starting states. Evolve them all by the same number of generations, until they have all stabilised. You now have a probability distribution over the final states. It's entropy is defined to be the sum over all the possible states of -p*log(p), where p is the probability of that state.

The entropy (with essentially the same definition) is an important quantity in thermodynamics. So I'm curious to know what it is in Life (even though I don't think thermodynamics is useful here). I conjecture that for large tori it's proportional to the area, although I don't imagine that this will ever be possible to calculate. The small tori probably don't give us much information about how it would work for large tori, but it would still be interesting.

Majestas32
Posts: 524
Joined: November 20th, 2017, 12:22 pm
Location: 'Merica

### Re: Small Tori in B3/S23

Also for 7x7 it's possible to omit Tori that are found by the evolution of other tori

dvgrn
Moderator
Posts: 6302
Joined: May 17th, 2009, 11:00 pm
Contact:

### Re: Small Tori in B3/S23

Majestas32 wrote:Also for 7x7 it's possible to omit Tori that are found by the evolution of other tori
If you keep a hash table of previously-seen 7x7 patterns, you mean? That won't need a petabyte of RAM like the 8x8 case, but it's still going to get to be a painfully big table before it has enough entries in it to be useful as a shortcut device... won't it?

I'm worried that the overhead from keeping a hash table for this purpose will be greater than the time you save. It might be faster to just run each pattern to stability as efficiently as possible, and not worry about whether it has been seen before.

The same might even be true of rotations/reflections/translations, unless there's a really quick way of enumerating only the unique 7x7 tori. (?)

Macbi
Posts: 714
Joined: March 29th, 2009, 4:58 am

### Re: Small Tori in B3/S23

Would it help to look at the tori which wrap with an offset? Then all the cells would be in
dvgrn wrote:The same might even be true of rotations/reflections/translations, unless there's a really quick way of enumerating only the unique 7x7 tori.
For circular strings there's an algorithm for generating only the canonical forms: https://en.wikipedia.org/wiki/Lyndon_word. Perhaps this could be adapted to the toroidal case (forgetting about the reflections)? Alternatively we could look at tori which wrap with an offset of 1, so that all the cells are in one long spiral path.

EDIT: Never mind, Lyndon words aren't quite what we want.

F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### Small Tori in B3/S23: First results

Dear all,
in the following posts I will present my first results for quadratic tori of size 2 to 5. Size 6 is in the making, 68 billion initial patterns take a while ... but the computational work is mostly done .

For each torus size ('tSize', from 2 to 5, until now) I have analysed all patterns (2^(tSize^2)) and their outcomes ('ash analysis'):
• * how many pixel in the ash
* which periodicity (incl p1, still lifes)
* how long it took to get there (generations from the start to either still life or end of one full oscillator cycle
and then the same analysis but for the 'canonical' initial patterns only.

These figures can answer the following questions (among others):
• - What is the longest period of any oscillator existing in this torus?
- What is the number of primary still lifes (all / canonical)?
- How many of the initial patterns end up 'empty'?
I am fully aware that there may be many other ways to analyse the B3/S23 behaviour of these tori. I will tackle the entropy suggestion as soon as I have finished the work on the 6 x 6 torus.

Again, any suggestions/feedback very welcome.

-Frank

F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### 2 x 2 Torus in B3/S23

Code: Select all

Torus Size: 2x2
# of cells: 4
# of states (raw): 16
# of states (canonical): 6

Start torus: 0
Finish torus: 15

[Start] 2018-05-10 16:36:48.478
[Finish] 2018-05-10 16:36:48.479

### ALL PATTERNS analysis: ###
Patterns analyzed:16
Generations analyzed: 11
Canonical patterns found: 6

ashPixCount|0:12|2:4|Total:16
ashPeriodCount|1:16|Total:16
ashGenCount|0:5|1:11|Total:16

Mean # of generations to 'ash': 0.6875

### CANONICAL ONLY analysis: ###
Patterns analyzed:6
Generations analyzed: 4

ashPixCount|0:5|2:1|Total:6
ashPeriodCount|1:6|Total:6
ashGenCount|0:2|1:4|Total:6

Mean # of generations to 'ash': 0.666667

Canonical Ash Patterns (CAP, n=2):

(0|p:1)
00
00

(1|p:1)
10
10


So indeed there is a stable non-empty form on the 2x2 torus.
-F

F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### 3 x 3 Torus in B3/S23

Code: Select all

Torus Size: 3x3
# of cells: 9
# of states (raw): 512
# of states (canonical): 26

Start torus: 0
Finish torus: 511

[Start] 2018-05-10 17:31:27.191
[Finish] 2018-05-10 17:31:27.195

### ALL PATTERNS analysis: ###
Patterns analyzed:512
Generations analyzed: 469
Canonical patterns found: 26

ashPixCount|0:386|4:126|Total:512
ashPeriodCount|1:512|Total:512
ashGenCount|0:127|1:301|2:84|Total:512

Mean # of generations to 'ash': 0.916016

### CANONICAL ONLY analysis: ###
Patterns analyzed:26
Generations analyzed: 24

ashPixCount|0:21|4:5|Total:26
ashPeriodCount|1:26|Total:26
ashGenCount|0:6|1:16|2:4|Total:26

Mean # of generations to 'ash': 0.923077

Canonical Ash Patterns (CAP, n=6):

(#0|px:0|period:1)
000
000
000

(#1|px:4|period:1)
010
110
010

(#2|px:4|period:1)
000
110
110

(#3|px:4|period:1)
010
110
100

(#4|px:4|period:1)
010
101
010

(#5|px:4|period:1)
010
011
100

No big surprises. No oscillator on 3x3, and 5 different (canonical) ash patterns (plus the empty one).
-F

F_rank
Posts: 29
Joined: March 28th, 2018, 6:01 pm
Location: Frankfurt, Germany

### 4 x 4 Torus in B3/S23

Code: Select all

Torus Size: 4x4
# of cells: 16
# of states (raw): 65536
# of states (canonical): 805

Start torus: 0
Finish torus: 65535

[Start] 2018-05-10 16:43:34.346
[Finish] 2018-05-10 16:43:35.105

### ALL PATTERNS analysis: ###
Patterns analyzed:65536
Generations analyzed: 228315
Canonical patterns found: 805

ashPixCount|0:49116|3:1696|4:7464|5:4736|6:320|7:384|8:1820|Total:65536
ashPeriodCount|1:56328|2:3896|4:64|8:5248|Total:65536
ashGenCount|0:53|1:10587|2:17424|3:15168|4:8320|5:2912|6:2816|7:2880|8:1536|9:1280|10:640|11:896|12:896|13:128|Total:65536

Mean # of generations to 'ash': 3.48381

### CANONICAL ONLY analysis: ###
Patterns analyzed:805
Generations analyzed: 2434

ashPixCount|0:617|3:14|4:91|5:40|6:3|7:3|8:37|Total:805
ashPeriodCount|1:714|2:46|4:1|8:44|Total:805
ashGenCount|0:6|1:177|2:227|3:179|4:92|5:31|6:25|7:25|8:13|9:10|10:5|11:7|12:7|13:1|Total:805

Mean # of generations to 'ash': 3.0236

Canonical Ash Patterns (CAP, n=23):

(#0|px:0|period:1)
0000
0000
0000
0000

(#1|px:3|period:2)
0000
0010
0010
0010

(#2|px:4|period:1)
0000
0110
0110
0000

(#3|px:4|period:2)
0100
0000
0011
0100

(#4|px:8|period:2)
0110
0110
0110
0110

(#5|px:6|period:4)
0010
0110
0110
0010

(#6|px:5|period:8)
0000
0110
0010
0110

(#7|px:4|period:1)
0010
0000
0010
0101

(#8|px:4|period:1)
0010
0001
1000
0100

(#9|px:5|period:8)
0101
0101
0010
0000

(#10|px:8|period:1)
0101
0101
0101
0101

(#11|px:4|period:8)
0010
1100
0010
0000

(#12|px:5|period:8)
0010
1110
0010
0000

(#13|px:5|period:8)
0000
1010
0010
0110

(#14|px:5|period:2)
0010
1010
1000
0100

(#15|px:6|period:8)
0010
1000
1010
0110

(#16|px:7|period:8)
0110
1100
1010
0100

(#17|px:7|period:8)
0010
1110
0110
1000

(#18|px:8|period:2)
0011
1001
0011
1001

(#19|px:4|period:2)
0100
0001
1000
0010

(#20|px:7|period:2)
0100
0111
1010
0010

(#21|px:8|period:2)
0100
0111
1110
0010

(#22|px:8|period:1)
0110
1001
0110
1001

The 4 x 4 torus already offers a few surprises. Not only does it have enough space for a blinker (no surprise), but there are many more p2's and even two different p8's (here shown on a grid of 9 tori of tSize=4):
'L dot':

Code: Select all

x = 12, y = 11, rule = B3/S23:T12,12
3bo3bo3bo$bo3bo3bo$2ob3ob3obo2$3bo3bo3bo$bo3bo3bo$2ob3ob3obo2$3bo3bo3b
o$bo3bo3bo$2ob3ob3obo!
and the other one:
'Small u':

Code: Select all

x = 11, y = 10, rule = B3/S23:T12,12
obobobobobo$3ob3ob3o3$obobobobobo$3ob3ob3o3$obobobobobo$3ob3ob3o! F_rank Posts: 29 Joined: March 28th, 2018, 6:01 pm Location: Frankfurt, Germany ### 5 x 5 torus in B3/S23 And last but not least here the analysis of the 5x5 torus: Code: Select all Torus Size: 5x5 # of cells: 25 # of states (raw): 33554432 # of states (canonical): 172112 Start torus: 0 Finish torus: 33554431 [Start] 2018-05-10 17:01:18.401 [Finish] 2018-05-10 17:05:30.620 ### ALL PATTERNS analysis: ### Patterns analyzed:33554432 Generations analyzed: 274631706 Canonical patterns found: 172112 ashPixCount|0:23487902|3:1324600|4:2385825|5:520110|6:3680900|7:11300|8:1338400|9:9500|10:86970|11:108700|12:600025|13:200|Total:33554432 ashPeriodCount|1:31115032|2:1491800|3:5600|4:733400|5:7300|10:30500|20:170800|Total:33554432 ashGenCount|0:3456|1:1135056|2:5484445|3:4038370|4:2280835|5:2164860|6:2242160|7:1990300|8:1749600|9:1677550|10:1444600|11:1402400|12:1035400|13:880100|14:780500|15:688300|16:586900|17:502600|18:477200|19:385700|20:410100|21:407000|22:269500|23:220300|24:181600|25:165200|26:126000|27:146800|28:142000|29:115200|30:91600|31:57800|32:48400|33:40400|34:32800|35:22400|36:27000|37:20400|38:26800|39:11000|40:7600|41:4400|42:3800|43:3000|44:2400|45:2000|46:2400|47:5600|48:6200|49:2400|50:1600|51:400|Total:33554432 Mean # of generations to 'ash': 8.18466 ### CANONICAL ONLY analysis: ### Patterns analyzed:172112 Generations analyzed: 1391774 ashPixCount|0:120630|3:6690|4:12249|5:2634|6:18885|7:60|8:6859|9:48|10:445|11:557|12:3053|13:2|Total:172112 ashPeriodCount|1:159754|2:7552|3:28|4:3723|5:40|10:161|20:854|Total:172112 ashGenCount|0:33|1:6289|2:28519|3:20853|4:11799|5:11183|6:11498|7:10186|8:8936|9:8544|10:7333|11:7089|12:5203|13:4416|14:3927|15:3456|16:2955|17:2526|18:2400|19:1938|20:2056|21:2037|22:1348|23:1102|24:908|25:826|26:630|27:734|28:710|29:576|30:458|31:289|32:242|33:202|34:164|35:112|36:135|37:102|38:134|39:55|40:38|41:22|42:19|43:15|44:12|45:10|46:12|47:28|48:31|49:12|50:8|51:2|Total:172112 Mean # of generations to 'ash': 8.08644  So, none of the 2^25 initial patterns needs more than 51 generations to either come to still life or have completed one full oscillating cycle. There are 172,112 canonical patterns. The mean number of generations has risen to just over 8 (from 3.5 for tSize=4). p2, p3, p4, p5, p10 and p20 oscillators exist. There are 33 'primary' canonical still lifes (including the empty one), as can be seen by ashGenCount|0:33. -F Majestas32 Posts: 524 Joined: November 20th, 2017, 12:22 pm Location: 'Merica ### Re: Small Tori in B3/S23 What are some of these p10 and p20 Tori? Please, stop spam searching Snowflakes. F_rank Posts: 29 Joined: March 28th, 2018, 6:01 pm Location: Frankfurt, Germany ### p10 / p20 on T5 in B3/S23 p20 on T5 is the standard glider, it needs two across-torus runs (each 10 generations) to be identically positioned. There are three different p10's on a T5, all shown on a grid of 3x3 tori (of tSize=5): p10 - 'A-U-A-U': Code: Select all x = 15, y = 14, rule = B3/S23:T15,15 2ob4ob4ob2o$bobo2bobo2bobo$2ob4ob4ob2o$bobo2bobo2bobo2$2ob4ob4ob2o$bob
o2bobo2bobo$2ob4ob4ob2o$bobo2bobo2bobo2$2ob4ob4ob2o$bobo2bobo2bobo$2ob 4ob4ob2o$bobo2bobo2bobo!

p10 - 'c/2 wall':

Code: Select all

x = 15, y = 13, rule = B3/S23:T15,15
3bo4bo4bo$2obob2obob2obo$15o3$3bo4bo4bo$2obob2obob2obo$15o3$3bo4bo4bo$2obob2obob2obo$15o!

p10 - still to be named:
I like this one a lot because it appears as if there is a pattern that moves faster than light speed (2 rows and 2 columns diagonally, in just two generations). Of course this an effect of the torus rules and no 'real' speed:

Code: Select all

x = 14, y = 15, rule = B3/S23:T15,15
obo2bobo2bobo$2obob2obob2obo2$o4bo4bo$b2o3b2o3b2o$obo2bobo2bobo$2obob 2obob2obo2$o4bo4bo$b2o3b2o3b2o$obo2bobo2bobo$2obob2obob2obo2$o4bo4bo\$b
2o3b2o3b2o!


Majestas32
Posts: 524
Joined: November 20th, 2017, 12:22 pm
Location: 'Merica

### Re: Small Tori in B3/S23

These are c/2, c/2, c diagonal agar's. Hmm

A for awesome
Posts: 1950
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1
Contact:

### Re: Small Tori in B3/S23

Majestas32 wrote:These are c/2, c/2, c diagonal agar's. Hmm
The last one is also c/4 diagonal in the other direction — possibly agarship potential?
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce