This result is really beautiful! At the same time, it's like GoL has been conquered, and in that it leaves me a little sad. But just a little bit. Congratulations!
I'd be surprised if there was a computable density limit for the random plane, just because the system is Turing complete so the answer could easily end up being like Chaitin's contant where it depends on halting problems. For example, note that an infinite random plane will contain, with probability 1, purely-by-chance prebuilt artificial intelligences with goals like "maximize number of blinkers in the plane". Though I guess it's also an open question if such a system could even survive and spread, as opposed to just getting eventually crushed by the surrounding noise.
Right, but I'm hoping it might be possible to prove that the limit exists (even if it's an uncomputable number), as opposed to some oscillating behaviour where the density infinitely often goes up to 70% and then back down to 30% for example.
This "conquering" is all just proof as to the fundamental nature of the Game of Life. It's not hard to imagine a working system of walking proteins and unzipping DNA structures in light of these findings. It really is beautiful.
I think the portion that is fundamental is deeper than game of life itself. Game of life is a rule set with sufficient complexity to get this far, but it isn't the only one. Anything with this class of behavior will support systems including those resembling DNA, the question is at what scale it emerges. If the scale is too big (arguably the scale for the result in this post is too big), it's a less elegant kind of emergence.
Agreed. One aspect of CA which interests me is robustness, the ability of a system to recover from error states.
What's been done with GoL is fascinating, but I think the next emergent layer of fascination for me is universal constructors which can handle a certain level of constant distributed noise or interference. A lot of times people just shrug and say, "well this pattern will always be critical/vulnerable in these locations, and cannot be made robust. But to me that opens up the door for entire classes of patterns which measure, embed and repair state of surrounding entities.
Very interesting. A biological cell is a 3-D self-replicating pattern (or is it 2-D rather?). Does GoL give us some insight into the nature of biological cells?
Definitely! It's hard to summarize that insight in any kind of concise way, though ... The Game of Life universe seems a bit too "fragile" to allow for the kind of emergent complexity that real-world physics supports. We can build self-constructing things like the https://conwaylife.com/wiki/0E0P_metacell , but if anything gets slightly out of place, the usual result is a truly horrific catastrophic explosion.
Conway's Life design work is kind of like building robots out of masses of subcritical uranium. Everything's fine until two robots unexpectedly bump into each other... which means you have to start out with everything very carefully balanced, such that that never happens.
So I guess one fairly obvious insight is that real-world physics supports more reliable and less explosive low-level structures than Conway's Life does, and those low-level structures can then safely be used as the basis for new levels of organization -- atoms -> molecules -> DNA -> bacteria -> eukaryotic cells -> multicellular organisms -> colonies of organisms -> ecosystems.
It's not clear how those higher levels of organization would work in Conway's Life. If they're possible, then they seem to be far beyond our current ability to simulate them -- though there's some recent research vaguely along these lines, about self-replicators that might be able to exert some control over the space around them:
* Is there an oscillator of every possible period? (We have them all except 19 and 41.)
* If you start off the entire plane in a random starting state, does its density tend to a limit as time goes to infinity?
* Is there a 'phoenix' oscillator, in which every live cell dies every generation, of period greater than 2?
* Can every pattern be destroyed by bombarding it with gliders?
* Is there an indestructible pattern?
At the moment people are working on building a spaceship which is only 1 cell tall in its starting state: https://conwaylife.com/forums/viewtopic.php?f=2&t=2040.