The image Mike has provided a link to is lovely and perhaps better (i.e., more informative) than what I asked for, but not quite what I wanted. One might stop the iterative construction at some finite stage, and plot the graph of the resulting piecewise-linear parametric curve in 3D. Indeed, some of you will rightly point out that such a graph is all I'm entitled to, given that any given image has only finite resolution. Then again, one can imagine an interactive program that allows one to dynamically zoom in on the graphical object by a factor of a googol or more; this interactivity would create as much of the illusion of infinite precision as a finite human being could ask for. Jim On Sunday, December 27, 2015, Mike Stay <metaweta@gmail.com> wrote:
http://www.shapeways.com/product/3MF7L6QKA/developing-hilbert-curve-large
On Sat, Dec 26, 2015 at 3:45 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
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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-- Mike Stay - metaweta@gmail.com <javascript:;> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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