[DesiLurker]

makes me wonder if its possible to get natural numbers like pi/e using a geometric structure in GoL. it would be interesting to derive them from an emerging order based on fixed set of automata rules. If possible it might lead credence to simulated universe hypothesis.

[terlisimo]

well...

1) GoL is turing complete

2) there are algorithms that calculate digits of Pi or e.

so... yes?

but if I just took any old Pi-digits algorithm and encoded it on GoL, its appearance would not be meaningful or "elegant" to our senses. You're probably asking "what does the shortest/most elegant program to calculate Pi in GoL look like, and does it maybe have some unexpected relation to other mathematical terms like, I dunno, Euler's identity or... Mandelbrot set?" And then you would probably need to answer the question "Well, how would you like the digits encoded and represented?".

All of a sudden your question becomes a bit ambiguous. Or did I misunderstand what you meant?

I mean.... I think I feel what you're asking, like... is there some primal version of Pi that can be encoded in GoL initial condition with as few bits as possible but I'm afraid that the answer is something like "well, that depends on what you mean by [...]"

[dvgrn]

Adam P. Goucher's pi-calculator pattern is quite elegant, I'd say. It prints the decimal digits in-universe, with Conway's Life blocks for pixels.

https://conwaylife.com/wiki/Pi_calculator

[DesiLurker]

yes you are right the question was ambiguous. WHat I was really looking for is if there is a way to physically derive PI out of some basic cellular automata operations. that does not necessarily have to involve circles etc. much less so about representing digits of PI itself. just that an pattern that evolves into ever more accurate values of pi (or a quantity derived from pi). this definitly must not be encoded or somehow preprogrammed into the GoL's initial condition. In fact I dont even care if the evolution rules of standard GoL are followed. just that SOME set of rules automatically produce PI-derivative.

the reason it would be cool is because then we've taken a most fundamental geometric constant and derived it from purely graph update type of mechanics. if you could do that then likely you can do a whole set of other things like physical laws.