The animation I intended was perhaps too computationally ambitious. Meanwhile, gosper.org/FHilbert.gif is just conventional plots of the first 288 approximations. I dislike such plots due to the finite line thickness, which someone might imagine contributed to spacefilling. If you read all the frames into Preview, say, you can see that the texture of frame n repeats at frame 4n. This gives us a way to interpolate the nonexistent frame n+1/4 from frame 4n+1. Also uploaded: gosper.org/outcirc.gif made by flipping a sign in incirc.gif. --rwg Stay tuned for a Ptolemaic Hilbert sweep from a multi-hour computation. On 2016-09-26 18:42, James Propp wrote:
I hope Bill will write a lively document (or create a lively video) that explains what's wrong with so many accounts of spacefilling curves.
(I myself wish that more accounts started in a "Not Knot"-ish vein, explaining why the "kindergartner's space-filling curve" --- scribbling one's crayon back and forth until the square is filled --- isn't a solution to the mathematician's problem.)
Is there a way to relax an approximation to a space-filling curve in continuous time so that it works out its kinks and regresses to simpler approximations?
(No interim self-intersections please!)
Jim Propp
http://gosper.org/FDrags128.mp4
If people really want to see it, Julian's recent Fourier matrix product
can
produce the analogous animation for Hilbert's "curve".
I would like to see it.
But it is a seductive thought crime to view a spacefilling function as some
kind of limit of "spacefilling curves"! Those curves Henry sketches are mere schematics, of no mathematical consequence. This leads inevitably to embarrassed hemming and hawing about how the area jumps from 0 to 1 at the very last moment, when both interior and exterior suddenly disappear and become boundary. Successive partial sums of the Fourier series are even more seductive. But no matter how many terms you take, you're still infinitely far from the end. --rwg
That's one way to look at it. But it depends on the notion of path-space you use and what metric you put on it, doesn't it? In particular, it's crucial to look at paths equipped with a parametrization. Then you really can get convergence to a limit. And the limiting object is a continuous function from a line segment onto a square. If you just look at the range of a path and not the parametrization, you can't describe the square as the limit in any meaningful way. Which I gather is part of Bill's point.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun