I have been looking at one-dimensional (1D) cellular automata (CAs) which generate two-dimensional (2D) fractal patterns. Among the 256 1D elementary CAs, I tried to list down the fractal generating ones by looking at pictures here https://www.wolframscience.com/nks/p54--more-cellular-automata/
I was wondering if it is possible to list down both elementary and non-elementary CAs which generate fractals? The elementary CAs that generate fractals can be listed down by checking the 256 patterns. But it also seems there is a formal result i.e. the elementary CAs which can be represented by a linear recursive relation for the CA states are the ones that generate fractals. I wonder if this is generalized to non-elementary CAs.
My main question is related to this and is simple to state: is there a fractal which can be embedded inside another fractal? Note that these two fractals do not have to be generated by elementary CAs. Any reference that goes into this will be helpful. By embedding, I mean that all the 1 automata states of one fractal coincide with a subset of the 1 automata states of the other fractal.