A note about categories of patterns that emulate various Elementary CA.
@dvgrn: It seems like you have most of this figured out, particularly by the time of
this post, but there still seems to be some confusion - not helped by the ConwayLife references to this subject being scattered about the place.
So firstly there are the bounded oscillators with XOR type behaviour. These are the 2x2 rectangular oscillators and the like, referenced earlier in this thread. They have very similar behaviour to bounded ECA, but I'm not sure that there's a direct correlation with any one ECA.
Then there are various line and block type oscillators which emulate one or other of the ECA. For bounded examples these correspond best when there are toroidal boundary conditions. These are the conditions which give rise to the various sequences of maximum oscillator periods on a circular grid of cells, e.g.
A085587 in OEIS (Rule 90 maximal oscillator periods). Here some examples with infinite lines:
Lines in CGoL - this emulates Rule 22 (and the same for lines in B3/S2). Kind of looks like XOR behaviour, but it's not.
Period = 24
Code: Select all
x = 1, y = 5, rule = B3/S23:T37,5
o$o$o$o$o!
Lines in B3/S25 - this emulates Rule 90. Classic XOR behaviour [ x_t+1 = x-1_t ^ x+1_t ].
A085587 A(37) = 87381
Code: Select all
x = 1, y = 5, rule = B3/S25:T37,5
o$o$o$o$o!
Lines in B3/S - this emulates Rule 18. Under certain conditions it behaves the same as Rule 90. Specifically: Grid length is a multiple of two and all on cells are a distance of 2m from every other On cell (for some integer m).
Period = 1022 (and it turns out this is closely related to
https://oeis.org/A268754 )
Code: Select all
x = 1, y = 5, rule = B3/S3:T37,5
o$o$o$o$o!
Lines in B3/S28 - this emulates Rule 150. The other XOR rule [ x_t+1 = (x-1_t ^ x+1_t) ^ x_t ].
A085588 A(37) = 29127
Code: Select all
x = 1, y = 5, rule = B3/S28:T37,5
o$o$o$o$o!
So the recently discovered phoenix agars emulate a subset of patterns which behave the same under Rule 90 and Rule 18. Those patterns are on a ring of even length, and all On cells must be either at odd or even sites along the ring.