On Wed, Oct 9, 2019 at 9:59 AM Bill Gosper <billgosper@gmail.com> wrote:
A couple of weeks ago, Julian found
Drag[(m + 1/2 + 1/4*(-1)^m)/2^n] == 1/2 (Drag[m/2^n] + Drag[(m + 1)/2^n]),
for positive integers m, n, where Drag(t) is the usual Dragon-filling map of [0,1] into the bounding box [-1/3 - i/3, 7/6 + 2i/3]. This describes where along the Dragon is the *m*th vertex of the (self-avoiding) *n*th order "median curve".
Using Julian's remarkable piecewiserecursivefractal interpolator-inverter applied to the Dragon's skin, as defined by http://larryriddle.agnesscott.org/ifs/heighway/heighwayBoundary.htm, I just noticed
Out[161]= ConditionalExpression[dragskin[1/3 - t] ==3/2 + I/2 - dragskin[1/3 + t], Abs[t] <= 1/6]
In[125]:= ConditionalExpression[I dragskin[1 - 3 t] == dragskin@t, 0 <= t <= 1/12]
In[126]:= ConditionalExpression[I ( dragskin[1/2 - t] - 1) -> dragskin[1/2 + t] - 1, Abs@t <= 1/4]
Plus an intriguing symmetry I haven't yet quantified, smooth scaling? <http://gosper.org/skin%20012o3.png>
Nothing so exotic. Just ConditionalExpression[dragskin[1/6 + t] == (1/2 + I/2) - 1/2 I dragskin[t], 0 <= t <= 1/4] It is remarkable that there are identities which scale the dependent variable (dragskin) by 2 and others that scale the independent variable t by 3. This is somewhat obfuscated (but not caused) by Riddle's(?) (wildly) nonuniform sampling density.
The whole boundary is dragskin([0,1]).
Unfortunately, Riddle's (really Mandelbrot's?) clever construction draws the skin at an unboundedly nonuniform rate. I.e., it has dimension D (an unlovely cubic surd between 1 (length) and 2 (area)), but its fill rate, measure(dragskin(t),t1<t<t2)/(t2-t1)^D, gets arbitrarily small. (And arbitrarily large?)
But these functional relations raise my hope of finding some continuous, monotone u(t) to give dragskin(u) a uniform fill rate.
—rwg
Mandelbrot once challenged me to find the skin dimension. I disbelieved the irreducible cubic from a 5⨉5 determinant. Mandelbrot wrote only terse expressions. So I found a 6⨉6. Same damn cubic. I called him up. "Yes, that's the cubic." "How the heck did you get that? it's not your style!" "I just looked at how the wiggles fit into each other."
It just occurs to me that I must have had two different, uniformly filling skin descriptions to write those determinants! Oh, to be young again.
I just remembered an ancient, Mil-Spec looseleaf notebook of XGP output including a uniformly sampled, hollow Twindragon, the source of at least one of those determinants. I may even have scanned it.