You can make the notion completely precise using aperiodic tilings.
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Firstly, I've realised that it's possible to elegantly convert any set of n Wang tiles (W_0, W_1, ..., W_{n-1}) into a 2D cellular automaton with An + B states (where A, B are some universal constants) where a single cell in state 1 attempts to be a spiral-growth pattern.
Specifically, if (1, c_1, c_2, ..., c_n) are the non-zero cells, in spiral order from the origin, then it attempts the following:
- If c_n represents a Wang tile W_k which interferes with earlier Wang tiles, increment its state to W_{k+1};
- If c_n represents a Wang tile W_n beyond the admissible range, delete it;
- If c_n represents a Wang tile W_k which does not interfere with earlier Wang tiles, append another cell representing W_0 to the end of the spiral.
If there is no admissible Wang tiling, this will terminate in finitely many steps. If there is an admissible Wang tiling, the state of the universe will converge to the lexicographically first (in spiral order) such Wang tiling, meaning that every cell will eventually be constant and represent the correct tile. (Proof: compactness/Tychonoff)
Note that this does not contradict the undecidability result, because it's undecidable to tell whether a tile has 'stabilised' or whether the spiral will at some point backtrack and delete that tile.
Note: I think it can be done with A = 3 and relatively small B, where a cell representing a tile carries an extra trit of information (testing head / valid head / non-head) and the few states contributing to the value of B are for determining which way to turn the snake so as to trace the spiral.
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For the second part of the correspondence, suppose you have a sequence of nested 2-state patterns, P0 <= P1 <= P2 <= P3 <= ... such that the limit of the P_i's (scaled to have constant width) is a self-similar fractal. Then you can take the limit of the unscaled P_i's to get an infinite pattern which, in a certain sense, 'corresponds' to that fractal in the same way that the limit of several cellular automata 'corresponds' to the Sierpinski triangle.
Now, if you can find a set of Wang tiles such that every P_i is locally derivable from every admissible tiling (and at least one admissible tiling exists!), then you can construct a nice rule table in Golly which will build the lexicographically first Wang tiling, in which you can see infinitely many copies of approximations to your fractal.
You could actually use Golly's 'icons' feature to make the resulting tiling, when zoomed in, look like one of the colourful illustrations of Wang tiles on Wikipedia. It would be fun to do this explicitly for the minimal aperiodic set of Wang tiles, after dealing with the annoying extra states required to bend around corners to achieve spiral growth.