I suspect it's what I want, if the third dimension is really the time-parameter along the curve (and not some sort of interpolated "development" parameter, as in the image Mike pointed me to). Unfortunately my access to my laptop is limited (for family reasons) and my iPhone isn't up to doing gnuplot images, so I won't know for sure until I get a chance to try it. In any case, thanks for creating the graphic; I'm sure it'll tell me something I don't know. Jim On Sunday, December 27, 2015, Joerg Arndt <arndt@jjj.de> wrote:
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 <javascript:;>> [Dec 27. 2015 07:48]:
On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de <javascript:;>> 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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