On Fri, Oct 21, 2016 at 7:25 PM, Bill Gosper <billgosper@gmail.com> wrote:
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
Julian's Fourier expander takes an optional argument to arrange m copies of a curve around a regular m-gon. gosper.org/6trifil.png . Instead of spacefilling an equilateral triangle divided into quarters, I tried dividing it into five little ones plus a double size one. Arranged around a hexagon, gosper.org/ringnoring1234.png (one of the few cases where the outline looks maybe better than the filled area.) (Shades of https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane ?) Note that with 2468 rotors, the Fourier approximation gets nowhere near the middle. So is this a spacefill, or not? Julian: "Of course it is ..." Proof that it reaches all three vertices of each triangle: In[422]:= hollow /@ {0, 4/5, 1} // Expand
Out[422]= {{0}, {3/2 + (3 I Sqrt[3])/2}, {3}} via (Julian's ever amazing) piecewiserecursivefractal Clear[hollow]; hollow[t_] := piecewiserecursivefractal[t, Identity, {Min[6, 1 + Floor[6*#]]} &, {6 # &, 6 # - 1 &, 3 - 6 # &, 6 # - 3 &, 6 # - 4 &, 6 - 6 # &}, {#/3 &, #/3 + 1 &, #/3/I^(2/3) + 3/2 + I Sqrt[3]/2 &, 3/2 + I Sqrt[3]/2 - #/3 &, 2 #/3 + 1/2 + I Sqrt[3]/2 &, #/3*I^(4/3) + 3 &}] A polygonal sampling: gosper.org/hollowfill150.png (Reminder: True spacefills self-contact like crazy.) --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.
[chop]