Small Tori in B3/S23

For general discussion about Conway's Game of Life.

6 x 6 Torus in B3/S23

This is my first attempt at the analysis of all possible patterns (2^36 = 68,719,476,736) on a 6x6 torus (under B3/S23 rules).

I have not done the CAP analysis that I have provided for the smaller tori. This may follow at a future timepoint with an improved software handling large (gigabyte-size) arrays.

`ashPeriodCountp    patternCount         2^36        pC [%]1   58,627,717,276   68,719,476,736   85.31%2    8,637,890,028   68,719,476,736   12.57%3            1,296   68,719,476,736    0.00%4      445,302,288   68,719,476,736    0.65%6       37,948,536   68,719,476,736    0.06%8        4,077,936   68,719,476,736    0.01%12     202,887,504   68,719,476,736    0.30%24     763,651,872   68,719,476,736    1.11%`

The full listing (AshPix, AshPeriod and AshGen) has been stored in my Dropbox and can be accessed via the following link:

Some observations:
- 63.8% of all initial patterns end up with an empty torus
- An additional 21.5% end up with a non-empty still life
- An additional 12.6% end up in a p2 oscillator
- That leaves 2.1% which end up in an oscillator >= p3
- p3 is rare (only 1,296 patterns end up in a p3)
- The longest period is p24, quite common actually (wait for more information on this channel)
- 4.5 bn (!) - 6.6% - of all initial patterns end up in a 12-cell ash pattern, whereas only 1.1 million (~ 0.025% of the previous figure) end up in an 11-cell ash pattern
- No ash pattern (incl. the oscillators) contains more than 18 cells (this threshold happens to be exactly 50% of all cells) (50% maximum density - coincidence? or is this a proven law in B3/S23?)
- No pattern needed more than 90 generations to determine its fate
- The mean number of generations (to ash) has risen to 11.6 (from 8.1 for Tori of tSize=5)
More to come.
-F
F_rank

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p12 / p24 on T6 (in B3/S23)

Not unexpectedly, p24 is a single glider on the 6x6 torus plane.

p12 is (mostly?) a group of c (orthogonal) agars (I think that is what they are called) - during my initial 16x16 experiments, I used to call these patterns "light speed walls".

At this point I cannot exclude the possibility of some rare forms of p12/p24 outside the above description.

-F
F_rank

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p3 on T6 (B3/S23)

This is a p3 on T6 (potentially, the only p3):
`x = 18, y = 18, rule = B3/S23:T18,18b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo\$b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo\$b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo!`
F_rank

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Re: 6 x 6 Torus in B3/S23

F_rank wrote:(this threshold happens to be exactly 50% of all cells) (50% maximum density - coincidence? or is this a proven law in B3/S23?)

It's proven that no stable (still life) agar can have more than 50% density; I'm not sure if there is a corresponding proofs for oscillating agars.

77topaz

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Re: p3 on T6 (B3/S23)

F_rank wrote:This is a p3 on T6 (potentially, the only p3):
`x = 18, y = 18, rule = B3/S23:T18,18b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo\$b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo\$b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo!`

Nice! That has a density greater than 50%.
`x = 81, y = 96, rule = LifeHistory58.2A\$58.2A3\$59.2A17.2A\$59.2A17.2A3\$79.2A\$79.2A2\$57.A\$56.A\$56.3A4\$27.A\$27.A.A\$27.2A21\$3.2A\$3.2A2.2A\$7.2A18\$7.2A\$7.2A2.2A\$11.2A11\$2A\$2A2.2A\$4.2A18\$4.2A\$4.2A2.2A\$8.2A!`
Gamedziner

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Re: p3 on T6 (B3/S23)

Gamedziner wrote:
F_rank wrote:This is a p3 on T6 (potentially, the only p3):
`x = 18, y = 18, rule = B3/S23:T18,18b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo\$b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo\$b4o2b4o2b4o\$2o4b2o4b2o\$3bobo3bobo3bobo\$2o2b4o2b4o2b2o\$b2o4b2o4b2o\$3bobo3bobo3bobo!`

Nice! That has a density greater than 50%.

No? 144/(18^2) = 44.4%

Macbi

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Re: p3 on T6 (B3/S23)

Macbi wrote:
Gamedziner wrote:Nice! That has a density greater than 50%.

No? 144/(18^2) = 44.4%

LifeViewer's stats show this agar's population oscillating between 177 and 180, with average density 178/324 = 53.94%
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Re: p3 on T6 (B3/S23)

BlinkerSpawn wrote:LifeViewer's stats show this agar's population oscillating between 177 and 180, with average density 178/324 = 53.94%

Hmm, so apparently we can't trust LifeViewer's population statistics for bounded grids (hit G to see them, for anyone who doesn't know this -- or hit H to see all the other optional displays you can turn on).

Notice that the population count stays appropriately at 5 for this case:

`x = 3, y = 3, rule = B3/S23bo\$2bo\$3o!`

-- but the count increases mysteriously whenever the glider runs across one of the bounds in this bounded-grid variant:

`x = 3, y = 3, rule = B3/S23:T18,18bo\$2bo\$3o!`

dvgrn
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Re: Small Tori in B3/S23

Yeah, and the birth and death counters behave weirdly too, even breaking the births - deaths = change in population rule.

77topaz

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Re: Small Tori in B3/S23

77topaz wrote:Yeah, and the birth and death counters behave weirdly too, even breaking the births - deaths = change in population rule.

Chris Rowett has a fix. It will show up with other recent fixes one of these days.

In the meantime, just be careful and don't believe the statistics if they say that an oscillator has a density over 1/2 --!

dvgrn
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Maximum density of small tori in B3/S23

I was wondering whether we can observe oscillators on small tori where at least one generation has a density > 0.5.
My results so far show that there is a 13-cell 'ash pattern' (these include all stages of oscillators) for a 5x5 torus and a 25-cell ash pattern for a 7x7 torus, so the answer is Yes. This leads immediately to the next very similar question: Is there any ash pattern with a density higher than (# of cells / 2) + 0.5?

Below you'll find the 13-cell pattern that oscillates on T5. It is a p10 that alternates between 13 and 8 cells, so the average density is 10.5 cells. More on the 25-cell oscillator on T7 later on this channel, stay tuned.
-F

`x = 5, y = 5, rule = B3/S23:T5,5bo\$obo\$o2b2o\$3obo\$2o2bo!`
F_rank

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p4 on T7 with a 25-cell generation

Here is the p4 on T7 that has one of its 4 generations with a density of just above 0.5 (25/49):

`x = 21, y = 21, rule = B3/S23:T21,212o3bob2o3bob2o3bo\$3b2obo3b2obo3b2obo\$3obob4obob4obobo\$2bobobo2bobobo2bobobo\$obobob2obobob2obobobo\$obobo2bobobo2bobobo\$ob2obobob2obobob2obo\$2o3bob2o3bob2o3bo\$3b2obo3b2obo3b2obo\$3obob4obob4obobo\$2bobobo2bobobo2bobobo\$obobob2obobob2obobobo\$obobo2bobobo2bobobo\$ob2obobob2obobob2obo\$2o3bob2o3bob2o3bo\$3b2obo3b2obo3b2obo\$3obob4obob4obobo\$2bobobo2bobobo2bobobo\$obobob2obobob2obobobo\$obobo2bobobo2bobobo\$ob2obobob2obobob2obo!`

The other 3 generations contain 22/23/24 live cells so the average density is 23.5 cells (i.e., below 0.5).
-F
F_rank

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Re: p4 on T7 with a 25-cell generation

F_rank wrote:Here is the p4 on T7 that has one of its 4 generations with a density of just above 0.5 (25/49):

Great! By packing the rotors more closely, you can get a density of 43/84:

`x = 56, y = 42, rule = LifeHistory:T56,42.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A\$.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A\$.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A\$.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A\$A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A\$4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A\$A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A\$2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A\$.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A\$.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A\$.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A\$A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.A\$2.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A\$2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A\$.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A\$.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A\$2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A\$A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.A\$A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A\$A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.3A\$A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A\$.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A\$.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A\$.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A\$.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A\$A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A\$4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A\$A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A\$2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A\$.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A\$.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A\$.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A\$A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.A\$2.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A\$2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A\$.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A\$.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A\$2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A\$A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.A\$A.A.A2.A.A.A.2A3.2A.A.A.A2.A.A.A2.A.A.A.2A3.2A.A.A.A\$A.A.2A.A2.A.A3.2A.A.A.A.4A.A.2A.A2.A.A3.2A.A.A.A.3A\$A.A2.A.2A.A.4A.A.A.A.2A3.A.A2.A.2A.A.4A.A.A.A.2A!`
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calcyman

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Re: Small Tori in B3/S23

Welp, we disproved that conjecture. RIP conjecture. Someone should update the wiki.

KittyTac

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Re: Small Tori in B3/S23

KittyTac wrote:Welp, we disproved that conjecture. RIP conjecture. Someone should update the wiki.

I'm pretty sure it's about average density, not the density of a single phase. Also, I thought it was proven; is it still a conjecture?
succ

blah

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Re: Small Tori in B3/S23

Yeah, none of those oscillators have an average density above 0.5, and wasn't that proven conjecture just about still lifes, anyway?

77topaz

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Representing objects in a toroidal universe

I will post my thoughts on the 'small tori' discussion from the apgsearch 4.0 thread to here.

calcyman wrote:I agree that this really needs to be a separate client program (with Catagolue as the server, of course, for simplicity).

At this stage, before anyone programs anything, it's worth discussing how best to represent objects in a toroidal universe.

Would it be worth differentiating between tori which we think can be explored exhaustively (in contrast to: 'based on a representative sample of all possible patterns') and those that seem beyond reach?

I don't know whether we have a consensus that - for the time being - everything beyond 8 x 8 is out of scope for an exhaustive exploration. I think 7 x 7 is clearly doable (with a joint effort, at least for the canonical patterns) and 8 x 8 is already a stretch.

For my work so far on 5x5, 6x6 and 7x7 I have used unsigned 64 bit integers as representation for any single pattern, so this works until 8x8. I have not used lifelib so far. Is there any introduction to lifelib? I have used shifting (horizontal and vertical), rotation and mirroring (only one axis needed) to reduce the number of all patterns to just the canonical patterns. For tori up to 8 x 8 I feel that it would not make sense to divide them into smaller objects that are detected/counted separately, I'd rather consider the entire 7x7 or 8x8 pattern (one generation - i.e., the 64 bit integer) as 'the' object. As an example, a single block, no matter where on the torus, would still be considered one single (canonical) object. All tori with two blocks would be regarded as different canonical objects as long as their relative position to each other is different. What sparks my interest in small tori are the rare and/or long period oscillators, the relationship between life expectancy (max no of generations) and torus size, the relationship between number of all possible patterns, canonical patterns, canonical ash patterns, Garden of Eden patterns, etc, all relative to the torus size. For these considerations it is of relatively little interest to count blocks or similar small island objects separately.

Tori greater than 8 x 8 are currently not my primary focus, but of course any conceptual considerations should take those into account as well. Perhaps the 'one pattern = one object' approach can be used for those torus sizes that we think can be explored exhaustively (imho up to 8x8), and anything larger than 8x8 could be approached in the more traditional way (based on random samples, with objects definitions as they currently exist, as far as possible).
F_rank

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Re: p10 / p20 on T5 in B3/S23

F_rank wrote: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:
`x = 14, y = 15, rule = B3/S23:T15,15obo2bobo2bobo\$2obob2obob2obo2\$o4bo4bo\$b2o3b2o3b2o\$obo2bobo2bobo\$2obob2obob2obo2\$o4bo4bo\$b2o3b2o3b2o\$obo2bobo2bobo\$2obob2obob2obo2\$o4bo4bo\$b2o3b2o3b2o!`

2 rows and 2 columns in 2 generations is just lightspeed diagonal.
I would like to name this "flies"
c(>^w^<c)~*
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Redstoneboi

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New results (mostly on 6x6 torus)

I have now updated my 6x6 results with the detection of the (initial) canonical patterns as well as the canonical ash patterns (CAP).

My software found the expected number of 239,123,150 initial canonical patterns (see http://oeis.org/A255016) out of 68.7 billion (2^36) initial patterns.
- 241 of these initial canonical patterns were stable from the beginning (ashGenCount=0), so we have 241 'unique'(=canonical) still lifes, including the empty 6x6 torus.
- This is confirmed by the same number (241) of CAP with ash.period = 1 (= stable). Always nice to show internal validity . The total number of CAP (for tSize 6) is 351 (quite a small number, actually, given the 239 million initial canonical patterns).

The following shows, for quadratic tori from size 2 to 6,
- the maximum number of generations (until stable or one full oscillating cycle completed)
- the number of canonical ash patterns (CAP)

For tori 2 to 6, these two parameters are in the same order of magnitude.

MaxNoGen_NoCAP_T2_to_T6_v2.png (6.56 KiB) Viewed 7366 times

The next chart shows, for a 6 x 6 torus, the number of 'still active' patterns on the logarithmic y axis, after n generations. n, on the x axis, goes from 0 to 90 = max# of generations for tSize 6.
So this starts with 2^36 (~ 68.7 billion) patterns in the upper left corner ('full' universe, no generations yet), and ends at 0 patterns (to allow for a logarithmic Y-axis, this one value has been manually changed from 0 to 1) after 90 generations in the lower right corner.
'Still active' is meant in the sense of 'of undetermined fate', i.e. not yet identified as stable or periodic.

NoPatternsStillActiveVsNoOfGen_T6_v2.png (14.02 KiB) Viewed 7366 times

More to come.
Frank
Last edited by F_rank on June 13th, 2018, 4:17 pm, edited 2 times in total.
F_rank

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Re: Small Tori in B3/S23

Just a quick note: on my system, those DropBox image links show up as broken-link icons in the actual message, but I can right click and open in a new browser tab with no problem.

If you want, you can upload those images as attachments to the message -- see the "Upload attachment" tab below the text area. After uploading, if you choose the "Place inline" option, the images will magically be displayed as part of the message.

(That's unlike other file types; things like ZIP files just show up as inline download links, but for PNGs and JPEGs and so on you should be able to see the image right away.)

dvgrn
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Re: Small Tori in B3/S23

Thanks a lot, done, including downsizing the original PNG files (to approx 660 x 360) to avoid scroll bars.
Frank
F_rank

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Re: Small Tori in B3/S23

That drop-off at precisely 90 generations is quite weird. What could the reason be for chaotic behaviour not being able to extend for more generations?

Because otherwise it would be similar to finding methuselae, where for a longer lifetime one would expect a (probably exponential) decay in the probability. However, in your graph, there is initially an exponential decay (with the exponential axis seen as linear) but then an incredible drop when close to 90 generations.
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Rhombic

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Re: Small Tori in B3/S23

Rhombic wrote:That drop-off at precisely 90 generations is quite weird. What could the reason be for chaotic behaviour not being able to extend for more generations?

Because otherwise it would be similar to finding methuselae, where for a longer lifetime one would expect a (probably exponential) decay in the probability. However, in your graph, there is initially an exponential decay (with the exponential axis seen as linear) but then an incredible drop when close to 90 generations.

This can be explained by the following sentence:
'Still active' is meant in the sense of 'of undetermined fate', i.e. not yet identified as stable or periodic.

If I was to count also the identified oscillating patterns as "still active", we would not see that drop off. I will respond in more detail later this week.

Frank
F_rank

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Re: Small Tori in B3/S23

A few questions about bounded-grid enumeration:
Has anyone attempted searches on grids other than unshifted tori? I'd imagine that there might be a similar amount of patterns on shifted tori and klein bottles.

Have FPGAs been considered for this type of search? You could easily do one generation per clock cycle at around 200Mhz. This would give about 9 hours to run each of the 2^36 possible 6x6 grids (without canonicalization) for 100 generations if it were completely sequential. Since this problem can be parallelized arbitrarily, many (probably >100) of the evolver units could be placed into the same large FPGA.
Hooloovoo

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6 x 6 Torus: analysis of canonical ash patterns

As written earlier, there are 351 canonical ash patterns on a 6x6 torus in the B3S23 world.

Here is the overview of the periodicity of these 351 patterns, and the respective frequency.
T6 CAP_PC table.PNG (8.42 KiB) Viewed 6601 times

Period 24 / count 2 is a simple glider in its two 'canonical' forms.
Period 12 / count 6 are actually two different oscillators.

(p12-1) is a 'light speed wall' oscillator with 4 canonical patterns (so 'canonical' periodicity is 4) that needs 3 full cycles to get to its origin (so period=12):
`x = 18, y = 18, rule = B3/S23:T18,183\$2b2o4b2o4b2o\$b4o2b4o2b4o\$18o4\$2b2o4b2o4b2o\$b4o2b4o2b4o\$18o4\$2b2o4b2o4b2o\$b4o2b4o2b4o\$18o!`

(p12-2) is also a 'light speed wall' oscillator with only two canonical patterns. It appears mirrored after traversing the 6x6 plane once which explains the period 12:
`x = 18, y = 18, rule = B3/S23:T18,182\$4bo5bo5bo2\$o4b2o4b2o4bo\$18o3\$4bo5bo5bo2\$o4b2o4b2o4bo\$18o3\$4bo5bo5bo2\$o4b2o4b2o4bo\$18o!`
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