[math-fun] Space-filling curves again

I just watched the "Space-Filling Curves" video on Numberphile (featuring Henry Segerman and his interpolating surfaces), and it rekindled my desire to see something similar that's a bit more canonical somehow. Is there a way to relax an approximation to a space-filling curve in continuous time so that it works out its kinks and regresses to simpler approximations? (No interim self-intersections please!) Jim Propp On Monday, December 28, 2015, James Propp <jamespropp@gmail.com> wrote:

Is there anything like this surface that has constant negative curvature?

That is: Can a topological disk embedded in 3-space with constant negative curvature have a (2-)space-filling curve as its boundary?