But any computer can only do approximations. Good enough to appear "the limit" to the human eye, though. So: choose a curve and level of approximation. Example: terdragon, iterate 8: Data: http://jjj.de/tmp-xmas/propp-terdragon.dat Script: http://jjj.de/tmp-xmas/propp-gnuplot.plt On you command line issue gnuplot propp-gnuplot.plt Whirl around with mouse. Is this (modulo perfection) what you meant? Best regards, jj * James Propp <jamespropp@gmail.com> [Dec 27. 2015 07:48]:
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?
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