I just 'invented' a closed plane-filling fractal that I don't recognize. I tried to Google for it, but I didn't have much luck with finding closed fractal curves on square grids. Anyway, I wrote up a blurb about it
on my website (where I have a bit more control over rendering than I do here).
Forgive the crappy ASCII art, but I wasn't willing to go to too much extended effort if it turns out that this is Horgledorf's Curve and everyone has known about it for ages.