On 8/19/10, Bill Gosper <billgosper@gmail.com> wrote:
Yesterday I told tutor Julian an idea for spacefilling a pentagon, and he said I could just conformally map Peano's filled square onto whatever, but then he got into it and helped make www.tweedledum.com/rwg/pentagonfill.pdf . It self-contacts, but the median curve doesn't. Note the density variations, which become unbounded in the limit. I don't have a fix, [...]
Now I do. http://gosper.org/pentfill.pdf for one of those exact fills, this time mapping Q(sqrt 5) into (onto?) Q(sqrt 5,sin pi/5,i) intersect <pentagon>. E.g., In[187]:= pentfill[1/2] Out[187]= 7/44 Sqrt[1/2 (85 - 31 Sqrt[5])] + 1/88 I (-57 + 49 Sqrt[5]) In[253]:= pentfill[(2 + Sqrt[5])^-3] Out[253]= -(1/22) Sqrt[5905 - 2602 Sqrt[5]] + 1/22 I (-133 + 52 Sqrt[5]) Even with adequate sense of left&right & dimension, I found this exercise difficult. I don't even see why it works, i.e., why should nested, unequal subinervals with surd endpoints be finite state? (And therefore permit finding fixed points.) Cautions: Note long runtimes! And with $RecursionLimit < 10^5, it is likely to quietly run forever, or maybe print batsh*t warnings. Also, polygonally joining images of sequences of abscissas will occasionally produce crossings, but these are sampling artifacts. --rwg