I have suggested the nomenclature “Z-function” and “Q-function” to distinguish between cases. Any Z-function depends on a choice of sampling domain, and there are infinitely many choices, some self-avoiding, others self-crossing, or self-intersecting. Q-functions, such as your “dragun”, are then the proper space-fillers, mapping surjective not injective, from Q to Q^2. Some of these computer drawings do remind me of Chen Rong “Nine Dragons”, where the dark, menacing forms are caught and depicted, somewhere between emptiness and filled space. —Brad

On Sep 18, 2019, at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:

It's a bit hard to get a self-avoiding polygon by periodically sampling the Dragon function. Here's one. <http://gosper.org/dragavoid.png> The usual recipe for a self-avoiding Dragon polygon is the "median curve <http://gosper.org/dragmed.png>", which, for fixed n, samples points m/2^n, but then averages pairs for adjacent m. Notice how this "median curve" (barely) stays inside the Dragon image. So for each n, there's a sequence of 2^n (unequally spaced) points in [0,1] that draw a median curve. Julian has found a remarkably simple formula <http://gosper.org/drag samp=med.png> (i.e., periodic sampling except for some phase modulation). This wasn't trivial—many median curve vertices have multiple preimages. Which are found by his function undrag, which of course inverts the function dragun. Self avoiding polygons are attractive because they're easy to follow, but I object to calling them "spacefilling curves", which anything but self avoid. They hit uncountably many points twice, and densely many thrice! —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun