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:
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...