Here's a raw sandbox with Peano and Wunderlich samplings: https://www.wolframcloud.com/obj/1fa6d29d-7451-4e68-ae7b-dded77390040 The "tetraskelion"s are dense with quadruple points. I need to figure out how, with Julian's tools, to make inversepeano and see if there are quintuple or sextuple points (which would be surprising). —rwg On Mon, Sep 23, 2019 at 6:45 PM Bill Gosper <billgosper@gmail.com> wrote:
On Sun, Sep 22, 2019 at 4:45 PM Bill Gosper <billgosper@gmail.com> wrote:
On 2019-09-22 06:15, Joerg Arndt wrote:
Even "Peano's curve" isn't unique: Walter Wunderlich: {\"{U}ber Peano-Kurven}, Elemente der Mathematik, vol.~28, no.~1, pp.~1-10, (1973). http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/ Don't let the German scare you, just check the images.
Wow, in http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf the 3⨉3 edge-traverser (Figur 5) is a perfect(ly confusing) hybrid of Peano's with Hilbert's! Gotta "Julianize" these. —rwg
E.g., ListLinePlot[ReIm[peano[#][[1]] & /@ Range[0/4/9/4/2, 1, 1/4/9/2/2]], Axes -> False, AspectRatio -> Automatic] tetraskelions <http://gosper.org/tetraskelions.png> (with indispensAble help from Julian. I have misunderstood his argument conventions all this time!) —rwg
For the n-dimensional version of (one of) the Peano curve(s): https://jjj.de/fxt/demo/comb/index.html#peano-ndim This is following A.\ J.\ Cole: {A note on space filling curves}, Software Practice and Experience, vol.~13, no.~12, (1983) and A.\ J.\ Cole: {A Note on Peano Polygons and Gray Codes}, International Journal of Computer Mathematics, vol.~18, no.~1, pp.~3-13, (1985). These papers win both my "horrible notation" and "most useless example" award.
Best regards, jj
P.S. regarding Peanistic curves: https://jjj.de/tmp-math-fun/all-R29-curves.pdf https://jjj.de/tmp-math-fun/all-R29-tiles.pdf View both files side-by-side.
* Bill Gosper <billgosper@gmail.com> [Sep 20. 2019 10:10]:
For decades I misapprehended Hilbert's spacefill as Peano's. Today at last, I corrected my web page: http://www.tweedledum.com/rwg/samhilbert.htm . The impetus was discovering that my blunder has been copied onto other websites. Unfortunately, with correct attribution.-} —rwg _______________________