Mandelbrot called this the "Snowflake Sweep" because it sweeps through every region of the Koch Snowflake.
This fractal curve is a natural self-avoider: it never touches itself no matter how many times it is fractalized. I made this video to introduce the wonderful world of fractal curves to everyone. If you like this video, chances are you'll love my website. And the book. And the other things I build and write about.