[math-fun] Ever seen a spacefilled pentagon?
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, but I can turn a slight variant of this one into one of those exact continuous maps of the rational unit interval onto some dense set of algebraic points. This would permit sampling at various frequencies and phases, which has proven interesting in previous cases (www.tweedledum.com/rwg/sampeano.htm, www.tweedledum.com/rwg/sam306090.htm ). --rwg
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
I added some pix and unpermuted some statements. Note the inadequacy of 105 digits in the insanely slow k/137 plot. I neglected to unwind-protect the cache, so whenever a pentfill call fails, you must Clear and redefine pentfill, trifill, and irtfill. See http://gosper.org/fst.pdf . You could get virtually identical pictures in a fraction of the time by punting the exact, fixed point cache in favor of returning .0 upon reaching a simple (< 1 pixel) recursion limit. --rwg Tutor Julian yawns that you can rather trivially spacefill polygons by triangulating them and then quadrisecting the triangles. On 8/31/10, Bill Gosper <billgosper@gmail.com> wrote:
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
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,[...]
It needn't even be conformal: http://gosper.org/tortilla.pdf --rwg ARTHROSCOPE PROTHORACES CRAPSHOOTER
If you're mapping a (filled) square onto a disc or whatever, why is there an extraneous solid black region at the bottom at the bottom of each drawing? The African masks are rather fine. WFL On 9/9/10, Bill Gosper <billgosper@gmail.com> wrote:
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,[...]
It needn't even be conformal: http://gosper.org/tortilla.pdf --rwg ARTHROSCOPE PROTHORACES CRAPSHOOTER
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