Is the similarity to a Dragon Curve (https://en.wikipedia.org/wiki/Dragon_curve) a coincidence? This post (https://cs.uwaterloo.ca/~csk/hat/) mentions a substitution system, which makes me think there might be a connection.
https://tilings.math.uni-bielefeld.de/substitution/penrose-r...
https://tilings.math.uni-bielefeld.de/substitution/fibonacci...
https://tilings.math.uni-bielefeld.de/substitution/semi-deta...
In addition, the dragon curve is a fractal -- mathematically, it's defined as the limit that the substitution process converges to as the details get "infinitely small", which means that in a sense, the true dragon curve (as opposed to the approximation that you can draw on a computer) has a boundary with no straight line segments at all. On the other hand, an aperiodic tiling is composed of finite, fixed-size tiles that extend outwards to infinity.
Funnily enough, the dragon curve is a space-filling curve that tiles the plane periodically.