Life cosmology

[Tropylium]

By now there is some good data on the natural frequency of various patterns. Has anyone been attempting to make something of this data? Explain it, somehow?

From what I've been looking at (tho this is still at the level of impressions, not of statistical data), the best single correlation to frequency seems to be the size of the smallest distinct precedessor. Assume clouds of activity that will randomly grow, shrink or split: small specific patterns are more likely to be extruded than large, and these will then do some final development into ash.

Achim F's results list does include numbers on smallest precedessors, but I don't see anywhere in there what these precedessors actually are. And in a few cases, it seems the pattern itself is counted rather than a distinct one.

Size is another factor as well. Once formed, a pulsar will obviously be more likely to be destroyed by a lingering reaction than, say, a longboat. However I'm not sure how exactly to quantize this. Surface area seems more relevant than density; another question is how to handle eating reactions in this context.

This model doesn't entirely work due to small methuselahs, however. I saw it noted in another topic that Herschels escaping near edges are the most common origin of ships from small seeds; similarly, I wonder how much of a contribution eg. centuries colliding with other junk have on the toad production frequency.

In more general, I would find it interesting to analyze what are the typical origins for any given object. For example, blinkers seem to mostly result from traffic lights, with a secondary contribution from these patterns I might call "blinkinserter 1" and "blinkinserter 2":

Code: Select all

..O..  .OOO.
OO.OO  O...O
.O.O.  OOOOO
..O..

(The first is the more common, and has several asymmetric 6-cell precedessors. )

Some patterns can also quite well emerge from chaos fully formed; this seems to be the case especially for snakes and *WSSes, which don't really have any considerably small precedessors. Blinkers and gliders do this commonly as well, but it seems that patterns with a width of more than 2 are rare to see form in this fashion. Which makes sense: clouds splitting usually involves a central overpopulated zone dying away, with only some edge cells and their possible descendants surviving.

Also it still has me stumped why pentadecathlons are as rare as they are. Pulsars have a handful of 7-cell precedessors, and yet they are almost 100 times more common. PDs have at least 144 possible 8-cell ancestors (from 2 traffic lights) which does not look that much worse. Is this then because two active components are required? I dunno; but maybe we could find out if we had some general stats here. FWIW, I've seen a natural PD occur twice (outside of mirror axes); one was indeed a 2-TL collision, the other a queen-bee-on-blinker reaction.

[Lewis]

I just did a quick experiment, finding the predecessors of blinkers (by hand) from small random soups. I only checked about 90 blinkers (it's tiring to backtrack the program and check each one by hand) and the results are as follows:

• 46 were formed by a traffic-lights predecessor that either survived intact or lost some blinkers along the way.
• 34 were formed from large random ash clouds that I had no idea how to categorise.
• 9 were formed from your pattern, 'blinkinserter 1'.
• 2 were formed from an unknown symmetrical predecessor shown below this list.
• 2 was formed from the Century methuselah.
• 1 Was formed from your pattern, 'blinkinserter 2'.

Here is the unknown predecessor mentioned above:

Code: Select all

x = 7, y = 5, rule = Life
bo$3o$4b3o$bo$5b2o!

I think a purpose-built program would be required to get any real results, especially about the formation of other objects (eg. what percentage of beehives are originally from honeyfarms etc.). Running each random soup and going back through it by hand takes far too long, even with small (20x20) soup sizes.

Another unrelated thing about oscillators' frequency - I always assumed high period oscillators (eg, the tumbler) would occur more commonly as they have more different states and therefore a higher chance of appearing than a p2 or still life of the same size. I don't know if it works at all though; most oscillators seem to have a few small predecessors and aren't 'born' in their finished state.

And I still can't understand how the clock is as rare as it is either, despite being as small as the toad and beacon (in one of it's generations).

[Extrementhusiast]

Another blinker predecessor to add:

Code: Select all

x = 7, y = 3, rule = B3/S23
bo3bo$ob3obo$2o3b2o!

[Tropylium]

I just did a quick experiment, finding the predecessors of blinkers (by hand) from small random soups. I only checked about 90 blinkers (it's tiring to backtrack the program and check each one by hand) and the results are as follows:

• 46 were formed by a traffic-lights predecessor that either survived intact or lost some blinkers along the way.
• 34 were formed from large random ash clouds that I had no idea how to categorise.
• 9 were formed from your pattern, 'blinkinserter 1'.
• 2 were formed from an unknown symmetrical predecessor shown below this list.
• 2 was formed from the Century methuselah.
• 1 Was formed from your pattern, 'blinkinserter 2'.

Here is the unknown predecessor mentioned above:

Code: Select all

x = 7, y = 5, rule = Life
bo$3o$4b3o$bo$5b2o!

Yes, that's exactly the kind of results I think would be interesting to see in larger amounts, including your discovery of that 10-blinker-maker.

For "large ash clouds", do you mean the blinkers emerged immediately from chaos? My method has been to note any precedessors that leave only the resulting object. I'd have expected the five 4-cell precedessors (or related 5-cell patterns with a domino or duplet spark on the left) to make the list too at least:

Code: Select all

O.O.  ..O.  ..O.  O.O.  ..O.
....  O...  ....  ...O  O..O
..OO  ..OO  O.OO  ..O.  ..O.

…tho my impression is indeed that they seem to be rarer than might be expected. (A variant of the last one is what makes an apppearence in century, at least.)

I also started looking at the precedessors of small seeds (∏, R, B, TL, HF, LoM) and the results are interesting. It does not seem the smallest precedessors in any way dominate ; for example, at least half of all pi-explosions actually result from an alternate path starting from eg. this 8-cell shape, which joins with the regular path in 3 generations (at the house stage):

Code: Select all

OOO
OOO
O.O

I'm also coming up with truckloads of HF precedessors in particular. Without much effort I've got 22 6-cell precedessors alone, which fall in 3 different groups that don't meet until the symmetric "large beehive" stage. For examples of one from each:

Code: Select all

OOO.OO  OO...  .O..
....O.  ...OO  O.OO
        O..O.  .OO.

I'm getting suspicions that having particularly small precedessors may actually be only an incidentally related factor to being frequent — these could both result from simply having a lot of small-ish (say, under 15 cells) precedessors.

I think a purpose-built program would be required to get any real results, especially about the formation of other objects (eg. what percentage of beehives are originally from honeyfarms etc.). Running each random soup and going back through it by hand takes far too long, even with small (20x20) soup sizes.

Right.

I've actually rigged a Life variant that uses a handful of extra states to flag the center of every blinker resulting from traffic lights. Something similar may be possible for honeyfarms. I'm not sure if it's 100% secure, but it's certainly quite context-specific.

Code: Select all

# TrafficLights
# marks blinkers arising from traffic lights
# state 0: OFF
# state 1: ON
# state 2: OFF via death by 4 neighbors or death of #3 by 5 neighbors
# state 3: ON from 2
# state 4: OFF from 8x#2 or death by 4 neighbors + pre-existing #4
# state 5: ON from near #4
# state 6: MARK
#
n_states:7
neighborhood:Moore
symmetries:permute

var a={0,2,4}
var b={0,2,4}
var c={0,2,4}
var d={0,2,4}
var e={0,2,4}
var f={0,2,4}
var g={1,3,5,6}
var h={1,3,5,6}
var i={1,3,5,6}
var j={1,3,5,6}
var k={0,1,2,3,4,5,6}
var l={0,1,2,3,4,5,6}
var m={0,1,2,3,4,5,6}
var n={0,1,2,3,4,5,6}
var o={0,1,2,3,4,5,6}
var p={0,1,2,3,4,5,6}
var q={0,1,2,3,4,5,6}
var r={0,1,2,3,4,5,6}
var s={0,1,2,3,4,5,6}
var t={1,3,5,6}
var u={1,3,5,6}

# Create state 4 (must override creation of state 2)
a,2,2,2,2,2,2,2,2,4
u,g,h,i,j,t,4,a,b,4

# Create state 5 (ditto)
e,h,i,j,4,a,b,c,d,5

# Create state 2 
1,g,h,i,j,a,b,c,d,2
3,g,h,i,j,t,a,b,c,2

# Create state 3
2,g,h,i,a,b,c,d,e,3

# Create state 6
1,h,5,a,b,c,d,e,f,6

# State 6 2-neighbor survival
6,h,i,a,b,c,d,e,f,6

# Normal 2-neighbour survival
g,h,i,a,b,c,d,e,f,1

# Normal 3-neighbour survival
g,h,i,j,a,b,c,d,e,1

# Normal 3-neighbour birth
f,g,h,i,a,b,c,d,e,1

# States die otherwise
k,l,m,n,r,o,p,q,r,0

Another unrelated thing about oscillators' frequency - I always assumed high period oscillators (eg, the tumbler) would occur more commonly as they have more different states and therefore a higher chance of appearing than a p2 or still life of the same size.

Seems about right to me. Pentadecathlon is more common than figure 8 is more common than mazing is more common than caterer, which are all about the same size. Symmetry may also play a role, tho…

And I still can't understand how the clock is as rare as it is either, despite being as small as the toad and beacon (in one of it's generations).

It has no nice small precedessors, however. There's an 8-cell one which is 4×5 (see the glider synthesis) but nothing smaller AFAIK.

By far most toads seem to rise from this pattern

Code: Select all

.OOO
O...
....
..OO

which can quite neatly arise from the edges of a pattern.

I've not looked at beacons yet in detail.

[Lewis]

For "large ash clouds", do you mean the blinkers emerged immediately from chaos?

From what I remember, the majority of them were formed as blinkers straight out of the chaos (ie. like most of the temporary blinkers in the evolution of the pi-heptomino). I'm sure several did form from smaller predecessors but I wasn't sure how to categorise them. Would the best way be to use the furthest-back predecessor that isn't affected by anything else (eg: T tetromino or whatever as opposed to traffic lights)?

I've actually rigged a Life variant that uses a handful of extra states to flag the center of every blinker resulting from traffic lights. Something similar may be possible for honeyfarms. I'm not sure if it's 100% secure, but it's certainly quite context-specific.

It seems to pick up most blinkers from traffic lights, the only ones it leaves tend to be ones that start from the filled 3x3 box. Apart from this I haven't seen it miss any.

Would there be any way to get the program to differentiate between the three types of HF predecessor?

[Extrementhusiast]

By far most toads seem to rise from this pattern

Code: Select all

.OOO
O...
....
..OO

which can quite neatly arise from the edges of a pattern.

I've not looked at beacons yet in detail.

Your pattern has an error. It should look like this:

Code: Select all

.OOO
O...
....
.OO.

[Dr. Monstaa]

By far most toads seem to rise from this pattern

Code: Select all

.OOO
O...
....
..OO

which can quite neatly arise from the edges of a pattern.

I've not looked at beacons yet in detail.

Your pattern has an error. It should look like this:

Code: Select all

.OOO
O...
....
.OO.

Just wondering that.... makes sense now