On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de> wrote:
Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke?
I'm not sure what constitutes a "stroke" in the continuum limit. A space-filling curve as a limit object is not a polygonal approximation or a sequence of such approximations; it's a continuous nowhere-differentiable function from [0,1] to the plane (constructed as the limit of such approximations). By way of comparison, consider the unit circle, parametrized at constant speed. The graph is {(t, cos t, sin t): t in [0, 2 pi]}. Projected onto the x,y plane, it's a circle; projected onto the t,x plane or the t,y plane, it's a sinusoidal arch. I'd like to see (among other things) the space-filling-curve analogues of those sinusoids. Something like Bolzano's everywhere-continuous-but-nowhere-differentiable function? Jim Propp