The Fourier series mentioned below is a specialization of the one in gosper.org/fst.pdf, which alternatively specializes to the Koch's Snowflake, Peano's spacefiller, etc., according to the parameter s. For convergence, the geometry of the recursion underlying the construction of the Fourier coefficients requires |s-i| < 2, but at s= √3, the infinite product telescopes, revealing that series convergence also extends to this boundary point. A couple of days ago I thought I noticed convergence beyond |s-i| = 2, but common sense and recent experiments at Julian's house convinced me I was hallucinating. In the process, we produced some pleasantly Picassoid squiggle <http://gosper.org/snowfourier.pdf>s. (Or maybe just Dr. Seuss.) The function futst(t,s,m,cutoff) traces the outline of the curve repeated around a regular m-gon as 0≤t≤2m, low-pass (t=time) filtered to the cutoff frequency (default ±69 harmonics). Thus lacking evidence of anomalous convergence, I was puzzled to receive just now (again, quoting without permission): You may have figured this out already, but just in case: the reason that it can converge while the fractal doesn't is that you found the series from an integral, but you don't actually compute the integral. Instead you compute a sequence of Riemann sums (intervals of size 2^-n), but this only necessarily computes the integral when the function in question is continuous. The fractal curve is not defined, and thus is not continuous, so the Fourier series is just some (necessarily discontinuous on a dense set, as otherwise you could find an interval where it would be continuous and match the "fractal" at dyadic rational values) function. It may or may not have to pass through the same points as the "fractal" at dyadic rational values. I don't know why it matches the predictions for 1/3, but the problem seems approachable. Julian He's apparently right! Out[219]= {futst[1/3, s, m, 9999], (I + s)/(3 + I s) + Cot[π/m]} In[227]:= %219 /. {9999 -> 99999, m -> 2., s -> 7/4.} Out[227]= {0.586944 - 0.00655471 I, 0.580311 - 0.00518135 I} where the Cot expression is the theoretical t=1/3 value for cutoff→∞. Trying 2 million terms, In[228]:= %219 /. {9999 -> 999999, m -> 2., s -> 7/4.} NEVER RETURNS, because somewhere between 99999 and 999999, Sum arbitrarily, silently, and PERVERSELY changes 1/2. from .5 to 1/2 ! This is infuriating. This is evil. --rwg Apropos Wikipedia math, I note that en.wikipedia.org/wiki/Generalized_continued_fraction is chatty, avuncular, and opinionated, and contains a URL apparently to the author's private web page, yet is free of all the usuak editorial unworthiness stigmata, as if it approached the pristine ideal of the breathlessly inconsequential http://en.wikipedia.org/wiki/MS_Smyril . Warren: For long range battleship type gunnery, the Ray solution would be useless because the atmosphere gets exponentially thinner at higher altitude which should be a major effect that he can't handle. In fact, to maximize range, they aim *above* 45°, the opposite of what you'd do with uniform drag [citation needed]. Henry: <minimum publishable quantum, fun facts> These are more than fun facts--they're valuable optimizations. We should gather up our various complex plane computational geometry slick tricks (circumcenter, circumradius, incenter, inradius, polygon area, polygon-point winding number, line-segment intersection, ...) and publish them somehow. Henry: (This is also true in the arts; take a look at Beethoven's manuscripts -- not everyone wrote music like Mozart, which were perfect the first time.) This is probably because Mozart wrote almost nothing in C#. Even less in C++. On Thu, May 24, 2012 at 10:54 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Thu, May 24, 2012 at 4:17 AM, Bill Gosper <billgosper@gmail.com> wrote:
I'm intrigued by Jack Holloway's recursive pentagram snowflake design, Fig 31 in the Minskys & Trinskys book<http://www.blurb.com/bookstore/detail/2172660>. However, circularizing and sampling obscure the pixel structure, and how, if at all, it relates to Fig 30. As usual, Julian comes to the rescue, with a "period 5" trinsky in which orbital x+y stays constant mod 2. The "black squares" (checkerboard) subset of this bitmap strongly resembles Fig 31, except the bounding pentagons have vertex angles of π/2 and 3π/4 instead of 3π/5. So I linearly crunched it to have "rhombic pixels" and wrote
http://www.tweedledum.com/rwg/rhombicp5.svg
The .svg is so you can scale it up without antialiasing. Unfortunately, my Apple Firefox scales it right off the screen, and refuses to scroll it back! Safari scrolls, but won't let you zoom very far. And Preview won't load svgs at all! And, as of this AM, gosper.org delivers source text instead of graphics for it. So to mega zoom it, Neil suggested downloading inkscape<http://inkscape.org/>, which seems to work great.
Can one speak of the limit of this pattern in the large? The pentagram outlines grow without bound and get "infinitely thin", yet seem to fill space. And what is a formula for their limiting shape?
Our Esteemed Moderator was so peeved by the stupidity of this question that, a couple of hours ago, he actually walked into my house, said
z[t]==Sum[((-1)^k*E^(I*(-(1/5) + k)*π*t)* Product[1 - Sqrt[5 - 2*Sqrt[5]]*Tan[((-(1/5) + k)*π)/(-2)^n],{n, ∞}])/(-(1/5) + k)^2, {k, -∞,∞}] 0≤t<10
and walked out. See http://gosper.org/fourierfracpent.png --rwg
Julian (quoted without permission) explains the original image as follows:
(d_t means the trinsky d multiplier, etc.)
It's the Trinsky equivalent of p5d1, which is d=g=1, e=2*sin^2(π/5). The Minsky-Trinsky equivalence is has d_t=g_t=d_m, e_t=e_m/2, has x about twice as large to compensate. This is what causes the checkerboarded images–for a given Trinsky (x,y), the equivalent Minsky point is ((x+⸤d_t*y⸥)/2,y) (sorry about the floors I haven't found any decent floor characters to use), and the checkerboard pattern comes from whether (x+⸤d_t*y⸥)/2 is an integer or not. The "checkerboard" pattern is only when the Trinksy d=g=1, in which case x+y is constant mod 2: x'=x-y=c mod 2, y'=y+⸤e*x⸥, x''=x'-y'=c-y' mod 2. A granularity of 1/2 would not produce checkerboarded images–it would produce four normal (i.e. at coordinates (ax+b,cy+d)) images (or two, alternating in either columns or row but not both, if only one granularity=1/2).
--rwg