[mvcalder]

For the experts in the house: would it be possible for this to wrap around, so the highest and lowest level join? A torus of GoLs? Can such a thing, assuming it exists, have a finite number of layers? Just curious, this is amazing.

[supriyo-biswas]

At a very high level, this is simulating the Game of life at the appropriate resolution level based on an algorithm whose input parameters are the zoom level and your location in the space, similar to a fractal pattern. So I'm not sure what you mean by a "torus" of GoLs.

[adrianmonk]

Have you heard of the thought experiment where a 2D plane is finitely sized instead of infinitely sized, but as you travel within the 2D plane, if you get to one edge, you wrap around to the opposite edge? Even though its size is finite, you can pick a direction and go that way forever.

If you sit in 3D space and look at this finite 2D plane, then it looks like (say) a rectangle. That's displeasing because from inside the plane, it's continuous, but from the outside, it looks discontinuous.

One way to get rid of that annoyance is to map the plane onto a torus. Then it looks continuous from the outside. It's no longer flat, which is un-plane-like, but you can't have everything in life.

Anyway, the zoom level of game of life is an infinite number line. But what if it repeats and can be represented as something that wraps around? You could think of it as a line segment that you wrap around into a circle (to join the two ends). So the same concept as mapping the plane onto a torus, except 1D instead of 2D.

[ordu]

Well... if we wrap around without meddling with time, it will be impossible. On the lowest level cells are switching much more frequently than on the highest level.

But if we allow ourselves to bend time, then probably... It would be like you zoom out and go back in time. My guess it is possible, but I still cannot wrap my head around infinite spatial dimensions of the field. We need infinity for that because without it will hit the problem of different cell count on the lowest level and the highest (which are represent the same sequences of states of GoL). I see no possible problems to buils such a "torus" but to be sure one needs to really prove it.

[dbdudbdiddjc]

What does that mean

[rpastuszak]

I guess what they mean is that the "zoom" would be cyclic, e.g. at the same time the game zoomed at 1x would look like the game zoomed at 101x, zoomed at 2x would look like 102x etc...

Totally different thing, but this makes me think of looping procedural animations, which are achieved by sampling noise on a circle (or walls of a cylinder in a 3d noise space).

[4ad]

It can't have a finite number of layers because of an argument very similar to the uncountability of reals.

[samatman]

To dispose of this briefly, the cardinality of computable functions is ℵ₀, and Life is a computable function. Although the parent question is underspecified (some good guesses as to what specifically is meant in sibling comments), no variation on that question could increase the ℵ number of the result.

[adrianmonk]

If there is a pattern that repeats, that pattern can have a finite length.

[4ad]

Such patterns are of measure zero due to the uncountability of the reals.

[adrianmonk]

I believe the comment above is talking about the game running as a simulation within another instance of the game. Each instance of the game has a state. Game G1 has state S1, game G2 has state S2, etc. I think they're asking if there's any S1 that can be chosen so that S2 = S1.

You might need 2 or more layers, so whatever number of layers you need (if this is possible, you'd have some sequence infinite sequence (moving through layers) like (S1, S1, S1, ...) or (S1, S2, S1, S2, ...) or (S1, S2, S3, S1, S2, S3, ...).

The length of the repeating pattern (subsequence) is going to be a positive integer, so I don't think real numbers are relevant, if that's the question.

[lostmsu]

So they still can exist. I don't think they meant that any such configuration would run, just one.

[4ad]

The empty board is a repeating pattern of length one.

[lostmsu]

The question was related to the simulation of game of life linked by the OP. E.g. Can a full simulation of game of life simulate itself.

[svara]

Yes but the question is whether there are repeating non empty patterns. I don't think you've answered this? I don't know but I think that's a fascinating question.