On the other hand, if you only look at a finite subset of the infinite field, then you find that knowing the exact contents of a n by n box in one generation only tells you the exact contents of an (n-2) by (n-2) box in the next generation. You have 2^(n^2) patterns mapping to 2^((n-2)^2) patterns, the former is 16^(n-1) times as large as the latter. This makes the existence of convergent patterns trivial, and the existence of Garden of Eden patterns quite surprising.
I agree with the GoE part, but does this really single-handedly imply convergent patterns? Two n×n states that produce the same (n-2)×(n-2) successor don’t necessarily have the same effects on their boundaries. Contrapositively, the part about only determining a (n-k)×(n-k) successor applies to any cellular automata that use a (k+1)×(k+1) neighborhood, even reversible ones.
I agree with the GoE part, but does this really single-handedly imply convergent patterns? Two n×n states that produce the same (n-2)×(n-2) successor don’t necessarily have the same effects on their boundaries. Contrapositively, the part about only determining a (n-k)×(n-k) successor applies to any cellular automata that use a (k+1)×(k+1) neighborhood, even reversible ones.
This is correct.
Thanks for pointing that out.