Contradicting my earlier email, I now compute the center (of mass) as simply ½ + *i*√3/6. (For the curve centered above the real axis, joining 0+0*i* to 1+0*i*.) flowsnake triple points of the form n/6/7/7/7 <http://gosper.org/flotripscirc.png> . This is exactly ⅓ + 0*i* from the image of ½, i.e. ⅙ + *i*√3/6, which is nowhere near the center. (Again I marvel at being able to evaluate at ½ a function originally defined only at "heptadic rationals".) With Julian's machinery, it should also be possible to produce the exact description of the boundary. Since it's continuous, it should provide another approach to the inradius and circumradius measurements, which are apparently numerically exotic. —rwg On Sat, Mar 21, 2020 at 9:03 PM Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Mar 19, 2020 at 5:40 PM Bill Gosper <billgosper@gmail.com> wrote:
I was chagrinned to realize that the popularity of this curve is probably due to how carefully it self-avoids when you sample it at the natural frequencies of 7^n. Ironically, like all true spacefillers, it's actually dense with triple points:
In[299]:= unflow[3/7 + I/7/\[Sqrt]3]
Out[299]= {5/42, 11/42, 17/42}
In[296]:= FlowS /@ {5/42, 11/42, 17/42}
Out[296]= {3/7 + I/(7 Sqrt[3]), 3/7 + I/(7 Sqrt[3]), 3/7 + I/(7 Sqrt[3])}
In[300]:= unflow[223/686 - (19 I)/(686 Sqrt[3])]
Out[300]= {257/2058, 263/2058, 281/2058}
In[297]:= FlowS /@ {257/2058, 263/2058, 281/2058}
Out[297]= {223/686 - (19 I)/(686 Sqrt[3]), 223/686 - (19 I)/(686 Sqrt[3]), 223/686 - (19 I)/(686 Sqrt[3])}
(Without Julian bailing me out!) I finally found a bug that was driving me nuts. <http://gosper.org/flotripscirc.png> I believe all the triple points are of the form n/6/7^k. —rwg