[math-fun] That curve I regrettably named flowsnake
Mandelbrot detested the name. Then he discovered a spacefill of Koch's Snowflake. The worthy claimant to the Flowsnake name! Which he still detested. I recently realized how little I know about my own curve, which fills the FranceFlake, the fundamental region for the positional number system Base: 2 + i^(2/3) = √7 e^(i arccos(11/14)/2) and the (necessarily) seven digits 0 and the 6th roots of 1. If FranceFill(0)= 0+0i, FranceFill(1) = 1 + 0i: 1) What is the area of a FranceFlake? 2) Where is its centroid? 3) Is its circumradius √(52/147) ? 4) Ironically, it was popular partly for how completely its canonical sampling (at 0/7, 1/7,...,1) self-avoided. But like all true spacefills, it is *dense* with *triple* points. E.g., what are the preimages of (9 + i √3)/21? —rwg
On Sat, Apr 6, 2019 at 9:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Mandelbrot detested the name. Then he discovered a spacefill of Koch's Snowflake. The worthy claimant to the Flowsnake name! Which he still detested. I recently realized how little I know about my own curve, which fills the FranceFlake, the fundamental region for the positional number system Base: 2 + i^(2/3) = √7 e^(i arccos(11/14)/2) and the (necessarily) seven digits 0 and the 6th roots of 1.
SPOILERS
If FranceFill(0)= 0+0i, FranceFill(1) = 1 + 0i: 1) What is the area of a FranceFlake?
Only √3/2.
2) Where is its centroid?
1/2+i/2/√3
3) Is its circumradius √(52/147) ?
No. ≥ √(6140317/3)/2401. (What the heck is this approaching?) The same as the FranceFlake, duh.
4) Ironically, it was popular partly for how completely its canonical sampling (at 0/7, 1/7,...,1) self-avoided. But like all true spacefills, it is *dense* with *triple* points. E.g., what are the preimages of (9 + i √3)/21?
{5/42, 11/42, 17/42} Are there quadruple points? How can we prove "No."?
—rwg
Pics or it didn't happen! Joking aside: I'd like to understand, but don't. Best regards, jj * Bill Gosper <billgosper@gmail.com> [Apr 08. 2019 15:50]:
On Sat, Apr 6, 2019 at 9:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Mandelbrot detested the name. Then he discovered a spacefill of Koch's Snowflake. The worthy claimant to the Flowsnake name! Which he still detested. I recently realized how little I know about my own curve, which fills the FranceFlake, the fundamental region for the positional number system Base: 2 + i^(2/3) = √7 e^(i arccos(11/14)/2) and the (necessarily) seven digits 0 and the 6th roots of 1.
SPOILERS
If FranceFill(0)= 0+0i, FranceFill(1) = 1 + 0i: 1) What is the area of a FranceFlake?
Only √3/2.
2) Where is its centroid?
1/2+i/2/√3
3) Is its circumradius √(52/147) ?
No. ≥ √(6140317/3)/2401. (What the heck is this approaching?) The same as the FranceFlake, duh.
4) Ironically, it was popular partly for how completely its canonical sampling (at 0/7, 1/7,...,1) self-avoided. But like all true spacefills, it is *dense* with *triple* points. E.g., what are the preimages of (9 + i √3)/21?
{5/42, 11/42, 17/42} Are there quadruple points? How can we prove "No."?
—rwg
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