This is a resend. Friggingidiot xmission software embargoed the original because I neglected to remove Julian's pretty inline pic, even though it was going to delete it anyway! On Wed, Sep 24, 2014 at 1:21 AM, Bill Gosper <billgosper@gmail.com> wrote:

* Bill Gosper <billgosper@gmail.com> [Sep 23. 2014 11:25]:

* Bill Gosper <billgosper@gmail.com> [Sep 22. 2014 15:55]: [...]

That's a beautiful clump of three rep-13s. But it's not obviously a clump of 13 stars.

jj>It should be (I could create it as a complex numeration system), but time is up for now; have to postpone it. Indeed _all_ my "tiles" should correspond to a numeration system, being rep-tiles, being doable as an IFS.

rwg>Julian sent a possible recipe that might not be tested before you leave. If it works, Betsy Ross can thank her lucky stars no Revolutionary thought of it in her time. Julian also sent an amazing codeberg that threatens to collect your t-shirt, if the offer survives dimacs.

jj>Great, offer will stand. I have a student exploring ways to present these curves in a setting like math-museum; the "amoeba flow" would be a very welcome addition (we thought of it, but I found no time so far and she doesn't want to program). Best, jj

--rwg

Lest there remain any doubt: ---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh@gmail.com> Date: Tue, Sep 23, 2014 at 2:36 PM Subject: Re: [math-fun] Jörg's t-shirt rep13 To: Bill Gosper <billgosper@gmail.com>

L-system for the boundary: axiom F+F+F+F+F+F+, rewrite rule F->F-F-F-F+F+F+F, angle=π/3 (this took me way too long to figure out). I wanted this because it lets me make nicer-looking 1-into-13s than the base-system method, because I can use Polygon rather than Point. Here's how I generated the attached (not, obviously, all in one go; it took a little debugging):

Lsys2fns["F", "F-F-F-F+F+F+F", \[Pi]/3`30]

Outer[#1[#2[#3[#4[0]]]] &, #, #, #, #] &[%];

Join @@ ((Function[L, L*#] /@ Exp[I*\[Pi]*Range[0, 5/3, 1/3]]) &@Flatten[% - (7/2 - (I Sqrt[3])/2)]);

Outer[Plus, (9/2 - 5*I*Sqrt[3]/2)*N[Join[{0}, Exp[I*\[Pi]*Range[0, 5]/3], 2*Exp[I*\[Pi]*Range[0, 5]/3]], 30], %];

Polygon[ri /@ #] & /@ %;

Graphics[Transpose[{Prepend[Hue /@ Accumulate[RandomReal[{1/6, 1/3}, 12]], Black], %}], ImageSize -> 1000]

gosper.org/13-flake order 5.png <http://gosper.org/13-flake%20order%205.png> You'll notice a couple of magic constants; they are ComplexExpand[z0 /. (Solve[(y - z0)/(x - z0) == w, z0] /. {x -> 0, y -> 1 - 2*I*Sqrt[3], w -> Exp[I*\[Pi]/3]})[[1]]] and (1-2*I*Sqrt[3])*jj13[1]/jj13[7/12], where 1-2*I*Sqrt[3] is the last point in the fractal generated by the L-system (with axiom F), jj13 is the original (non-flake) fractal, and jj13[7/12] is the point where three of them touch when making a flake.

Julian

On Tue, Sep 23, 2014 at 12:32 PM, Bill Gosper <billgosper@gmail.com> wrote:

...

On Tue, Sep 23, 2014 at 2:01 AM, Bill Gosper <billgosper@gmail.com> wrote:

...

* Bill Gosper <billgosper@gmail.com> [Sep 22. 2014 15:55]:

jj>Extra points if you do that gif-ery for the curve http://jjj.de/tmp-ryde/t-shirt-R13-15-hires-cropped.png (L-system at bottom, turns are by 120 degrees). Total curve should be light gray, and the moving part should be 1/13 of the total (so the same shape will appear 13 times, as indicated in the image). ... plus you get a T-shirt! (If you use the image to, well, print one. I got one and it looks quite OK).

Best, jj

That highly symmetric (almost dihedral 6) supersnowflake from three of these in a loop--is that also rep13? Yours?

I assume you mean page 10 of http://jjj.de/tmp-rwg/all-r13-tiles.pdf Here is generation 4 of the same: http://jjj.de/tmp-rwg/r13-15-plus-tile-4.pdf

It has rotational 6-symmetry, which (apparently) can only happen for orders of the form 6*k+1. No flip-symmetry (so indeed not D_6 as you mention), and I am fairly sure none of my curves has D_6.

Mine? I'd think so.

Note I'll be leaving tomorrow morning (that's in 24 hours from now) for http://dimacs.rutgers.edu/Workshops/OEIS/announcement.html and will be away from my mail until 12.10.

Best, jj

That's a beautiful clump of three rep-13s. But it's not obviously a clump of 13 stars. Julian sent a possible recipe that might not be tested before you leave. If it works, Betsy Ross can thank her lucky stars no Revolutionary thought of it in her time. Julian also sent an amazing codeberg that threatens to collect your t-shirt, if the offer survives dimacs.

--rwg

Here's Julian's evidence that it is indeed rep13: gosper.org/jjstar.png (site just froze--can't test) To get Julian's original solid fill, {}->{0}. Why wasn't this found earlier? Mandelbrot would have amygdalated with envy. It easily rivals the Kochflake and Franceflake, and will soon(?) have a large Googleprint. --rwg

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