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I'm not sure whether the idea of a spaceship can be made to work on such graphs (at least if the ruleset only cares about the multiset of states of cells N1(u) and N2(u) but not on the ordered pair of cellstates).
Suppose there is some compact configuration with only finitely many "alive" cells ...

I wrote an explanation of what I did here: https://github.com/GeethanPfeifer/sierp ... /caohg.pdf

confocaloid wrote: January 9th, 2025, 1:12 am ...

Cool, thanks.

That sounds interesting. Implementations, anyone? (It looks unclear)

It's nearly five years late, but I managed to get this idea running in Golly.
You can see a demonstration of it below in confocaloid's post. (Or, watch https://www.youtube.com/watch?v=0USVznP6Id8 though it's blurry.)
You can ...

The main difficulty I see with this is defining what a cell would be. In a Sierpinski triangle, each triangle-cell indeed only has three numbers, but how do you "isolate" a triangle to simulate it as a cell in a CA? Every sub-triangle of a Sierpinski triangle is divided into more triangles without ...

Has anyone looked into cellular automaton on fractals, such as the Sierpinski triangle?
It would be interesting to investigate this branch of types of cellular automaton.

On a side note, the sierpinski triangle is an interesting example in of itself as each cell only has itself and three neighbours ...