Four Dragon curves generating outwards from the same vertex will traverse every edge of a grid exactly once (and as a consequence will be plane-tiling as well).
I am captivated by this fact, and somewhat wishfully suspect that there is a simple and illuminating explanation for it. If this is not the case, however, can anybody direct me to a resource where this is discussed in detail?...I cannot find the original articles by Chandler and Donald J. Knuth. "Number representations and dragon curves", on jstor.org :(
Here are a few quick links for reference:
http://mathworld.wolfram.com/DragonCurve.html
https://en.wikipedia.org/wiki/Dragon_curve
Historically, Dragon Curves were discovered in 1969 by two NASA engineers who were interested in the pattern produced by folding a piece of paper repeatedly and then unfurling it such that all the folds were manifested as right angles, a fundamental construction which I believe to be part of the key to intuitively understanding all these properties...
Here’s the substitution rule that generates the dragon curve. It replaces each arrow with two smaller rotated arrows with particular orientations.
If we apply the same substitution rule to this infinite grid of arrows, we happen to get a smaller rotated copy of the same grid. After seeing it happen once, it’s obvious that it will continue to happen infinitely many times.
Now we can color four arrows of the initial grid and watch them grow into four dragons:
We’ve generated exactly the figure that you posted. But this construction makes it clear why they have to fit together this way—each arrow in the grid must be used exactly once.