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? Jim Propp 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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