[ojorjgmk] 1D Lego fractal

A one dimensional cellular automaton generated using the following state transition rules. Note that each cell in the next line (generation) depends on four parents, so it is best drawn on a hexagonal grid, rather than the square grid on which 1D cellular automata are usually drawn (three parents). Vaguely reminiscent of the Sierpinski gasket, and more to Rule 30.

* represents either 0 or 1 ("don't care"), so 16 rules total, as one would expect with 4 parents.

*00* -> 0
*11* -> 0
101* -> 0
*101 -> 0
001* -> 1
*100 -> 1

The general idea is that cells grow to the left and right, but try to avoid crowding. The structure is a binary tree: branches never re-meet.

Bigger picture (click to enlarge):

What is the asymptotic density?

Inspired by an attempt to build a dense Lego pyramid that could still be easily taken apart. This fractal is the vertical cross section; concentric squares around the vertical axis. (The inner structures turned out to be too fragile.)

Haskell source code to generate the images.