Did *no-one* try this?? In any case, Howard has added Julian's sextuple point triangle filler: http://hichacks.is-a-geek.net/triangles/?a_=sexttri&radiusPct=90&nRotors=111... A so-far undiagnosed numerics issue limits it to 1261 rotors--barely enough to suggest the 25 2nd level sextuple points. As a static, unfilled curve, here it is with 111 and 6251 rotors: gosper.org/sexpt.png Here is a url for the Gibbs triangle I mentioned: *http://hichacks.is-a-geek.net/triangles/?a_=gibbstri&radiusPct=85&nRotors=11... <http://hichacks.is-a-geek.net/triangles/?a_=gibbstri&radiusPct=85&nRotors=111&dtPct=20&rotorSort=1&cmn_=32&cmt_=highcontrast&cmabs_=false&stepsPerCycle_=1&delayMS_=0>* Here's Frequency Overmodulation: *http://hichacks.is-a-geek.net/triangles/?a_=besser&radiusPct=85&nRotors=111&... <http://hichacks.is-a-geek.net/triangles/?a_=besser&radiusPct=85&nRotors=111&dtPct=20&rotorSort=1&=4&cmn_=32&cmt_=highcontrast&cmabs_=false&stepsPerCycle_=1&delayMS_=0>* Here's five times worse: *http://hichacks.is-a-geek.net/triangles/?a_=besser&radiusPct=85&nRotors=111&... <http://hichacks.is-a-geek.net/triangles/?a_=besser&radiusPct=85&nRotors=111&dtPct=20&rotorSort=1&=20&cmn_=32&cmt_=highcontrast&cmabs_=false&stepsPerCycle_=1&delayMS_=0>* *These all perform nicely even without Chrome.* --rwg Speaking of spacefiller sextuple points, there's the "tree with backtracking" base 2 spacefill, Figure 14 in gosper.org/Article.pdf, the gasket paper I mentioned yesterday. I wonder if Julian's recursive fractal magic applies. And whether it extends to finding symbolic expressions, given recursions containing parameters. Note that Cesaro's halfsquare fill ( https://en.wikipedia.org/wiki/De_Rham_curve), is both a tree with backtracking and a Koch construction, so Julian's stuff applies. On Thu, Feb 23, 2017 at 6:11 PM, Bill Gosper <billgosper@gmail.com> wrote:

On 2016-08-31 16:55, Bill Gosper wrote:

Hi Vitaliy, I have a sandbox with a general Fourier animator. It takes a formula for the

coefficients (amplitudes and phases as functions of frequency), speed, and coloration.

It isn't very fast because it colors one pixel at a time, emulating the Symbolics

mathematically correct draw-triangle ALU-add microcode necessary for smoothly drawing

moving edges. Mathematically correct means that, if two triangles share an edge, there

will never be missing or overwritten pixels, and if the triangle is "inside out", the "bump"

will be negated. (The trick is to consistently define

exactly which pixels to bump.) With this primitive, you can then animate the movement

of an edge segment simply by drawing two triangles filling the quadrilateral defined by

the old and new positions of its endpoints. To draw a seamless T joint, just [include] the

"zero area" triangle formed by the constituent edges, which will usually add or subtract

the "bump" constant to or from a few pixels. --Bill

Howard Cannon has implemented a fast, full-res re-creation of the Symbolics triangle primitive,

along with several lovely Ptolemaic sweeps, plus luxurious controls:

http://hichacks.is-a-geek.net/triangles/artcircle

E.g., for a nice picture of 2D Gibbs ringing, change the "Circle" box to Gibbs Tri, and click >.

Besser illustrates frequency (over)modulation. The vector sum runs back and forth <Amplitude>

radians along the circumference of the circle

Howard (hic@iname.com) tailored the page for Chrome, and seeks your comments and suggestions.

(I'll try to supply him with the coefficients for Julian's sextuple point spacefill.)

--rwg