Still life analysis

[Tropylium]

This is going to be a topic for me to post various musings on still lifes and related stuff.

---

I'll start with something simple: chain still lifes, and the family of semilakes. The lakes, for those not yet aware, are a family of chain-shaped (in terms of connectivity) still lifes composed of dominoes, the smallest example being the pond. This can be generalized to some extent, however: any "tip" of a lake can be contracted from a domino to a monomino. This must be done in pairs to ensure connectivity remains. Still lifes generated in this fashion I call semilakes. The simplest two, corresponding to the pond, are the beehive and the tub.

For further demonstration, here are the four smallest lakes and all the corresponding semilakes:

Code: Select all

x = 97, y = 72, rule = B3/S23
4b2o4bo4bo$3bo2bo2bobo2bobo$3bo2bo2bobo3bo$4b2o4bo4$7b2o10bo10bo$6bo2b
o8bobo8bobo$6bo2bo8bobo8bobo$4b2o4b2o4b2o3b2o4b2o3b2o$3bo8bo2bo7bo2bo
7bo$3bo8bo2bo7bo3b2o3b2o$4b2o4b2o4b2o3b2o6bobo$6bo2bo8bobo8bobo$6bo2bo
8bobo9bo$7b2o10bo5$7b2o13bo15bo$6bo2bo11bobo13bobo$6bo2bo11bobo13bobo$
4b2o4b2o7b2o3b2o9b2o3b2o$3bo8bo5bo7bo7bo7bo$3bo8bo5bo7bo7bo5b2o$b2o7b
2o4b2o6b2o6b2o5bo$o8bo5bo7bo7bo7bo$o8bo5bo7bo8b2o3b2o$b2o4b2o7b2o3b2o
11bobo$3bo2bo11bobo13bobo$3bo2bo11bobo14bo$4b2o13bo5$7b2o10b2o10b2o10b
2o10b2o10b2o10b2o10b2o$6bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2b
o$6bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo$4b2o4b2o4b2o4b2o4b
2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o$3bo8bo2bo8bo2bo8bo2bo8b
o2bo8bo2bo8bo2bo8bo2bo8bo$3bo8bo3b2o4b2o4b2o6bo2bo8bo2bo6b2o4b2o4b2o4b
2o4b2o4b2o6bo$4b2o4b2o6bo2bo8bo3b2o4b2o4b2o4b2o3bo8bo2bo8bo2bo8bo3b2o$
6bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo6b2o4b2o4b2o3bo$6bo2bo6b2o4b2o4b2o
3bo6b2o4b2o4b2o4b2o4b2o4b2o3bo8bo2bo6b2o$4b2o4b2o3bo8bo2bo6b2o3bo8bo2b
o8bo2bo8bo2bo8bo2bo8bo$3bo8bo2bo8bo2bo8bo2bo8bo2bo8bo3b2o4b2o4b2o4b2o
4b2o4b2o$3bo8bo3b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o6bo2bo8bo2bo8bo2bo$4b2o
4b2o6bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo8bo2bo$6bo2bo8bo2bo8bo2bo8bo2b
o8bo2bo9b2o10b2o10b2o$6bo2bo9b2o10b2o10b2o10b2o$7b2o3$7bo11bo23bo23bo
11bo11bo$6bobo9bobo21bobo21bobo9bobo9bobo$6bobo9bobo21bobo21bobo9bobo
9bobo$4b2o3b2o5b2o3b2o17b2o3b2o17b2o3b2o5b2o3b2o5b2o3b2o$3bo7bo3bo7bo
15bo7bo15bo7bo3bo7bo3bo7bo$3bo7bo4b2o3b2o16bo7bo16b2o3b2o5b2o3b2o5b2o
5bo$4b2o3b2o7bobo19b2o3b2o19bobo9bobo9bo2b2o$6bobo9bobo10b2o9bobo10b2o
9bobo7b2o3b2o5b2o2bo$6bobo7b2o3b2o17b2o3b2o17b2o3b2o4bo7bo3bo5b2o$4b2o
3b2o4bo7bo15bo7bo15bo7bo3bo7bo3bo7bo$3bo7bo3bo7bo15bo7bo16b2o3b2o5b2o
3b2o5b2o3b2o$3bo7bo4b2o3b2o17b2o3b2o19bobo9bobo9bobo$4b2o3b2o7bobo21bo
bo21bobo9bobo9bobo$6bobo9bobo21bobo22bo11bo11bo$6bobo10bo23bo$7bo!

Note the two empty slots, showing that semilakes cannot be generated mechanically.

(The last lake is the third smallest only in terms of area; there are three others with 40 cells. See wikilink above.)

Not all chain-shaped still lifes are lakes or semilakes, tho. Two simple exceptions are the loaf and the mango. "Not-quite semilakes" also exist, that locally (within a radius of some 3 cells or so) look like they could be semilakes, but globally aren't; I think this is the smallest example:

Code: Select all

x = 22, y = 10, rule = B3/S23
4bo3bo3bo3b2o$3bobobobobobobo2bo$3bobobobobobobo2bo$b2o3bo3bo3bo4b2o$o
20bo$o20bo$b2o4b2o4b2o4b2o$3bo2bo2bo2bo2bo2bo$3bo2bo2bo2bo2bo2bo$4b2o
4b2o4b2o!

And then there's this kind of stuff:

Code: Select all

x = 31, y = 31, rule = B3/S23
16bo$15bobo$15bobo$9bo3b2o3b2o$8bobobo7bo$8bobobo7bo$6b2o3bo6b2o$5bo
11bo5b2o$5bo11bo4bo2bo$6b2o4bo5b2o2bobo$8bo2bobo6bobo2b3o$8bo2bobo6bob
o5bo$9b2o3b2o5bo5bo$16bo9bo$16bo10b3o$4b2o4bo3b2o14bo$3bo2bo2bobobo11b
5o$3bo2bo2bobobo10bo$b2o4b2o2bo2b2o9b3o$o10bo4bo11bo$b2o9b4o11bo$3bo
22bo$b2o9b6o7bo$o10bo6bo5bo$b2o4b2o2bo2b4o5bo$3bo2bo2bobobo8bo$3bo2bo
2bobobobo5bo$4b2o4bo3bobo3bo$16bo2bo$16bobo$17bo!

Open question #1: what's the third (fourth, fifth…) smallest chain still life that isn't a (semi)lake?

No, not the paperclip; that one has S3 cells on the long sides. An easy way to distinguish a chain still life is that it does not require S3 for stability, since every cell has exactly two neighbors. (Still lifes with multiple chains are also possible though, like the aircraft carrier, or a very long beehive stabilized by two normal beehives…)

Here are chains of population 21 and 22 which I suspect are optimal:

Code: Select all

x = 9, y = 21, rule = B3/S23
3b2o$2bo2bo$bo2bo$o4b3o$obo5bo$bobo3bo$3bo2bo$3bobo$4bo3$4bo$3bobo$2bo
2bo$bo3bobo$o5bobo$obo5bo$bobo3bo$3bo2bo$3bobo$4bo!

(Incidentally: the SW-by-NE fuses can be extended indefinitely, so there exist chain still lifes of any size > 20.)

[Tropylium]

—Another quick term: I don't think there is a collectiv name for the family of tub/barges/boats/ships yet? I've been calling them vessels.

(Sidenote: the block can be considered a length-0 ship. Or if counting the number of cell pairs rather than internal cells, length-1.)

[Tropylium]

Next, cracking down on some bigger stuff.

The biggest traditional division in the classification of still lifes has been set between pseudo and strict (non-pseudo) still lifes.

I am coming to the conclusion that one of the most natural grounds of classification of still lifes is thru consideration of their subsets of life cells, in particular subsets that form still lifes on their own. (More on this later.) This is alreddy apparent in the concept of pseudo still lifes: by definition, a stlife (< I'm going to be using this abbreviation from here on for brevity) is pseudo if its live cells can be partitioned into two or more sets, which are all still lifes on their own, and which are not connected to each other.

What has always struck me as unnecessary here is the requirement for unconnectedness. We don't care whether a "basic" still life is connected (eg. integral) or unconnected (eg. table on table). So why should this be a critical issue with pseudo stlifes, when partitionability alone works just fine, too, to produce all of them?

Now, the meaning of "pseudo" is ingrain'd by now and I'm not trying to change it… but I think a more general division would be into composite and non-composite stlifes, defined as above, just ignoring connectedness.

So this would mean there are composite stlifes whose components are connected. And indeed, they're already known as ties, and just like with simple pseudo stlifes, anyone can easily see how simple ties (shiptie, snaketie, etc.) decompose into smaller stlifes. It is this way in which I find they're pretty much on level with the concept of pseudo stlifes. We can obviously mix and match, too, having some components connected and others isolated (eg. block on shiptie).

---

Subclassification of composite stlifes is possible, too, and indeed this works pretty much the same for pseudo stlifes and ties. For an example, we know examples of pseudo stlifes which can be partitioned into 3 or 4 stlifes, but not fewer; this is entirely possible with ties, as well…
• A triple-tie: partitionable into a snake (in the center), an Eater 2-ish cross (in the upper right), and a fancy squiggle, but not into two stlifes:

Code: Select all

x = 13, y = 12, rule = B3/S23
8bo$7bobo$5b3ob3o$4bo7bo$3bob3ob3o$b3o3bobo$o3b2obo$2o2bob2obo$bo6bobo
$o7bobo$b3ob2obob2o$3b2ob2o!

(There are probably simpler examples…)
• A quad-tie; partitionable into two beehives and two blocks on tables, but not into two stlifes:

Code: Select all

x = 8, y = 10, rule = B3/S23
bo3b2o$obo2b2o$obo$bob4o$3bo2bo$bo2bo$b4obo$5bobo$b2o2bobo$b2o3bo!

(It can be partitioned into a two-beehive constellation and the two blocks on tables, though)

I suggest defining the rank of a composite stlife thus: considering a minimal partition, ie. which is not a refinement of another partition, the rank is the number of distinct components it can have at most. In this sense, the previous example of "triple-tie" has rank 3, a triple pseudo stlife will also have rank 3, etc.
…Or perhaps considering the entire set of minimal partitions would be better? A ship tie ship tie boat would then have a rank of (2, 2) rather than simply 2, for example.

Another statistic that may be useful is the largest number of components a stlife can be partitioned into, which I'm tentativly calling the multiplicity. This runs into some issues of uniqueness though. It is entirely possible to partition stlifes in conflicting ways, for example this "filled lake" into either a lake 2 and a central pond, or four ponds at each cardinal direction:

Code: Select all

x = 10, y = 10, rule = B3/S23
4b2o$3bo2bo$3bo2bo$b2ob2ob2o$o2bo2bo2bo$o2bo2bo2bo$b2ob2ob2o$3bo2bo$3b
o2bo$4b2o!

A set approach might again work here; this could be then said to have a multiplicity of (4, 2). The rank would be (2, 2, 2, 2, 2, 2, 2, 2), I think (the lake/pond partition, four one-pond/triple-pond partitions, two diagonal biponds partitions, and a partition into constellations of two opposite lakes).

(I'm open to adjustments of terminology, if you have better ideas.)

---

As an aside, I hypothetize the following four statements hold for still lifes of any given sufficiently large cell count (in any rule supporting an infinity of still lifes of these types, not just Life!):
— There are more pseudo than strict still lifes.
— There are more ties than non-ties.
— There are more strict ties than strict non-ties.
— There are more non-tie pseudo than non-tie strict still lifes.
(And obviously, also more composite than non-composite still lifes.)
The currently known stats for the number of pseudo vs. strict still lifes suggests this much, at least: while strict stlifes have a head start, pseudo stlifes have by 16 cells reached 50% as many in number, and by 24 cells, 75% as many. As the size grows, adding one cells seems to allow about 2.57 times as many pseudo stlifes, 2.45 times as many strict stlifes… so I'd expect pseudo stlifes to overtake in number by 30 cells (since 0.75·(2.57/2.45)⁶ > 1).

A presupposition I'm not going into yet is whether the number of still lifes of X cells (of any type) actually grows exponentially, or perhaps slightly faster due to the increase in area available for new cells…

Heuristically, this all seems obvious since that a large still life can easily have many small tie and pseudo components hanging around its edges; requiring these to be essential rather than detachable components would be a major limitation.

Whether strict ties or non-tie pseudo stlifes grow faster in number, I have presently no real idea; this would presumably have mainly to do with the number of ways a composite stlife of either kind may be achieved, which I'd currently guess suggests ties to take the lead?