20 ms·

Is the similarity to a Dragon Curve (https://en.wikipedia.org/wiki/Dragon_curve https://en.wikipedia.org/wiki/Dragon_curve) a coincidence? This post (https://cs

• teraflop 4y ago
  As far as I know, the resemblance is superficial. (For one thing, the standard dragon curve is based on 90° angles, and this tile has a mixture of 90° and 120° angles, giving it a 6-fold pseudo-symmetry.) There are many other non-periodic or aperiodic tilings that are based on substitution rules, and many of them look totally different: https://tilings.math.uni-bielefeld.de/substitution/penrose-rhomb/ https://tilings.math.uni-bielefeld.de/substitution/penrose-r... https://tilings.math.uni-bielefeld.de/substitution/fibonacci-times-fibonacci-variant/ https://tilings.math.uni-bielefeld.de/substitution/fibonacci... https://tilings.math.uni-bielefeld.de/substitution/semi-detached-house/ 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.