On 2015-12-27 22:36, Joerg Arndt wrote:
Nice one! My browser refuses to play the file, saying it is corrupt. So wget http://gosper.org/DDrag.mp4 and vlc DDrag.mp4 to the rescue.
Best regards, jj
Thank you for persisting! Ahah, http://gosper.org/TDrag3c.mp4 is a Fudgeflake filled by three Triadic Dragons, m=3 in http://gosper.org/fst.pdf , p144 (actual page 16) et seq. The Fourier coefficients are infinite products of ordinary scalars. A technique similar to Julian's amazing little piecewiserecursivefractal function permits exact evaluation at any rational time value. Plugging said time into the Fourier series and equating produces closed forms for sums of infinite products which I think would have even gotten a rise out of Ramanujan. Earlier in the paper this is exhibited for the Snowflake family of fractals, leading to the pair of identities (d247) and (d246) in http://www.tweedledum.com/rwg/idents.htm , where the only difference on the lhs is changing 260 to 261. It's a shame that our unfamiliarity with matrix products makes it hard to appreciate the analogous identities for the dyadic (Heighway) Dragon and Sierpinski Gasket. As explained in fst.pdf, the Snowflake and Terdragon Fourier series were also derived with matrix products, which then miraculously telescoped. --rwg
* rwg <rwg@sdf.org> [Dec 28. 2015 07:18]:
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.