# How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case "How many three-dimensional Hilbert curves are there?" http://www4.ncsu.edu/~njrose/pdfFiles/HilbertCurve.pdf but I need 2d.

Actually I'm looking for plane filling non-self-intersecting curves with short Lindenmayer representation. But I have no ideas how to generate all those except with brute-force. Is there any literature on the subject?

• What do you mean by a "plane filling non-self-intersecting curve"? The paper to which you gave a link does not include this notion. Oct 3, 2018 at 20:30
• I’m pretty sure plane filling curves can’t be bijective. Oct 3, 2018 at 20:44
• @ThomasRot I meant like Hilbert curve it can self-contact without self-crossing. Oct 3, 2018 at 20:57
• @WlodekKuperberg I would like curves which behave like Hilbert curve. Its limit fills the square while finite representation it is not self-intersecting and even limit case is self-contacting without self-crossing. Oct 3, 2018 at 21:27
• You probably need a definition of "self crossing". Oct 4, 2018 at 0:21