To the left is an image of the first 2048 rows in the evolution of a certain 2-state, 1-dimensional cellular automaton. The initial condition consists of a single black cell on a white background. The state of a cell on each subsequent row depends on the states of four neighboring cells on the previous step specifically, this is rule number 39780.
This rule is one of 256 left bijective rules of this size, and the pattern it generates has fractal dimension log4(5). I discuss it in a paper on local nestedness and positional bijectivity in cellular automata.
Unlike many rules that exhibit global nested structure, this rule is not additive. One indication of this is that its width increases like √t, rather than linearly, with time.