Could there be an app that lets you draw a freehand closed curve, then internally computes the Fourier transform, and then draws your curve on the screen using epicycles? Jim Propp On Thursday, September 8, 2016, Bill Gosper <billgosper@gmail.com> wrote:
Jörg nailed it. This is similar to http://gosper.org/TDrag3c.mp4 (plays in Safari but not Firefox). Gawd, I can't believe E. H. Moore would perpetrate such an inelegant kludge! Gluing four Hilbert arcs onto the sides of a degenerate rhombus apparently for the mere purpose of creating a closed squarefilling curve. Whose quadrants are not Moore curves! Better would have been filling a domino with two Hilberts (back to back) on the sides of a digon. gosper.org/fst.pdf explains how to get the Fourier series of a curve to repeat on the sides of a regular n-gon, indented for negative n. http://gosper.org/TDrag3c.mp4 is three triadic dragons on the sides of an equilateral triangle.
Actually, for Fourier purposes, the curve need not even close. It can just snap back after each period, interpolating just one point halfway through the snap. Better yet, why didn't "algoritmic" draw a https://en.wikipedia.org/wiki/Sierpiński_curve, which is closed and legitimately four copies of itself.
Apparently the "disks" are to emphasize that the rotors (harmonics) retain fixed amplitudes. But they don't assure the skeptical viewer that their relative frequencies and phases remain fixed.
I wonder if "algoritmic" had an analytic formula for the nth Hilbert harmonic. Given some such formula describing an arc, I'll bet there's a neat formula for arranging them on the sides of an arbitrary rhombus.
I'm not surprised there seem to be excess rotors in the video. Spacefilling Fourier series converge very slowly, You typically need to double the number of harmonics for each additional level of detail of the spacefiller.
And, it would help if "algoritmic" started and ended with zero angular velocity, like http://gosper.org/TDrag3c.mp4 . --rwg
On 2016-09-08 16:30, Joerg Arndt wrote:
* James Propp <jamespropp@gmail.com <javascript:;>> [Sep 08. 2016 17:59]:
Can Bill Gosper or anyone else explain what's going on with https://twitter.com/algoritmic/status/772699702064254976 ("Moore curve drawn with epicycles")?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Unless I am very mistaken this is more and more terms of the Fourier series (in polar coordinates). The disks seem to indicate the magnitude of the terms of the series (but there seem to be too many disks for my feeling after the first go-around is finished).
One of the many many cases where people really should indicate what they are doing when putting pictures/movies online.
Best regards, jj
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