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? 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