See gosper.org/julianstrifle.png . Note the tiny self-crossings not visible in the cursive drawing. It is usually far clearer to portray a <2-dimensional curve as the boundary of a two-dimensional region. --rwg On Thu, Oct 20, 2016 at 4:49 AM, Bill Gosper <billgosper@gmail.com> wrote:
On 2016-10-19 10:11, James Propp wrote:
Very pretty. What's the math here? (Apologies if you've already answered this question.)
Jim Propp
Julian wrote a nifty Fourier expander for recursive Koch polygons. E.g., ptsgnlst2Fouriermat[{0, 1, I^(2/3), 1 + I^(2/3), 2}, {1, -1, 1, -1}] spacefills the triangle joining 0 to 2 via 1+i√3. Actually, it makes an infinite 3x3 matrix product for the coefficients a(k). Then Sum a(k+1/m) exp(2 i π (k+1/m)) repeats the fractal on the sides of an m-gon. The gif just accumulates consecutive harmonics. --rwg
On Wednesday, October 19, 2016, Bill Gosper <billgosper@gmail.com>
wrote:
With much help from Julian, gosper.org/hellodoily.gif --rwg
On Fri, Sep 30, 2016 at 10:13 AM, Bill Gosper <billgosper@gmail.com <javascript:;>> wrote:
On Thu, Sep 29, 2016 at 8:33 PM, Bill Gosper <billgosper@gmail.com <javascript:;>> wrote:
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.
Three hours. Foo, the Symbolics machine could do this nearly in real time. gosper.org/hilbert286.gif is the 287 rotor sweep. (288 would have been too gross.) It was to have been a single frame of the frequency expansion animation, analogous to gosper.org/FHilbert.gif , but the project would not have contributed much insight into the final "catastrophic" dimension bump.
I haven't tried very hard to telescope Julian's 3⨉3 matrix product for the Fourier coefficients. Success would provide a flood of strange identities like (d247) and (d246) in http://www.tweedledum.com/rwg/idents.htm , which come from the family that includes the Sierpinski Curve and Koch's Snowflake (http://gosper.org/fst.pdf pp137, 138) rather than Hilbert's curve. Failure to telescope would encumber the infinite products in the identities with matrices instead of scalars. --rwg The infinite Fourier series faithfully represents Hilbert's function, except it misses the top two corners (the endpoints of Hilbert's curve) and moves them instead to top center (where the animation begins and ends). This is a quadruple point if you think of the Fourier function on [0,1], but only a triple point if you regard it as a periodic function, since a period contains only one of its endpoints.
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