On 2015-12-27 00:15, Dan Asimov wrote:
This sounded very appealing, but now I wonder if such a thing would be so stretched-out in 3D along the R^1 direction that the only interesting view of it might be *along* the R^1 direction, in which case you'd just see the usual graph in the plane.
Still, this seems definitely worth trying.
* * *
Is there a nice sequence of self-avoiding curves — approaching an area-filling curve — that are defined by simple Fourier series?
http://gosper.org/DDrag.mp4 But the only formula I found for the coefficients is an infinite product of 3x3 matrices. (There's a dimension parameter, with self-avoidance for D<2.) --rwg
I also wonder if there are some particularly nice geometric conditions that one might ask of such a sequence, such that there is essentially only one solution. (Maybe there's a way to make the idea of such curves being "simple" rigorous.)
I'd guess there's a better chance of finding such a thing if instead of the usual n-cube, the codomain of the curve were instead a cubical n-torus R^n / Z^n.
—Dan
On Dec 26, 2015, at 7:42 AM, James Propp <jamespropp@gmail.com> wrote:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.