Bill, That curvy Snowflake-filler is nice! Here's something I wrote aboutJörgArndt's solution for filling the Koch snowflake, where the segments of the curve all have the same length: https://spacefillingcurves.wordpress.com/2019/09/01/on-filling-the-koch-snow... -j ‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐ On Friday, September 4, 2020 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? 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