On Fri, Sep 4, 2020 at 9:38 PM Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Sep 3, 2020 at 4:56 PM Bill Gosper <billgosper@gmail.com> wrote:
What the heck is going on here? Alex's (fill rate) smoother idea on Mandelbrot's
Snowflake filler: [Picture <http://gosper.org/smoothman9colors.png>, worth a thousand words, censored for bagbitting math-fun.]
somehow manages to fill both a Snowflake and one √3 times larger with the exact same pattern. Mandelbrot missed it, along with everybody else. I never suspected it was even possible. What's the rule? Note that successive sizes are "texturally everted". —Bill
Julian to the rescue. <http://gosper.org/smoothflake.png> Oh what I'd give for BBM's reaction to this!
Speaking of bounding boxes|circumradii|convex hulls, recall APG's heroic extraction of a gross of digits of the circumradius of the France ("Flowsnake") Fractal. For which we never found an expression. Do we even have an accurate bounding box for the terDragon?
Apparently we didn't! I compute it to be {{-1/16, 17/16}, ±3√3 I/16} ={{Re[terDragon[1/240],Re[terDragon[239/240]]}, ±Im[terDragon[t]]} (another 239 factoid!) where t = precisely 113022934338503101/337661456635224360 = FromContinuedFraction@{0, 2, 1, 79, 3, 4, 3, 1, 1, 3, 2, 1, 1, 1, 1, 5905, 5, 2, 1, 2, 2, 3, 2, 74, 3, 2, 37, 1, 2} ! terDragon@t = 299201287/573956280 + 3 I √3/16 which is why I think I haven't slipped off the rails here. But it's admittedly hard to imagine why Im[terDragon] should be maximal at such an exotic abscissa. —rwg If you want to mess with this stuff, especially if you want to fact-check my crazy value of t, ask me to send you Julian's relevant magicware. I just approximated (minus) the left bound of Alex Roodman's i √7
spacefiller <http://gosper.org/smoothman.png> as FromContinuedFraction{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 5, 2, 94, 1, 1, 1, 42, 1, 2, 1, 2, 1, 1, 19, 23, 1, 2, 1, 2, 16, 55, 1, 14, 93, 1, 23, 1, 2}. Octupling the sampling frequency gave
{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 27, 2, 1, 2, 2, 2, 1, 2, 34, 12, 1, 8, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 24, 1, 3, 10, 2, 1, 1, 10, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 3, 3}
I see no sign of √7.
I always rave about piecewiserecursivefractal's inversion capability, but
I should caution that fractal functions like the (optionally) smooth flakefill have inverses in finite terms only on arguments of the form a + i b √3, where a and b are rational.
—rwg