[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.