[math-fun] Hilbert's ≠ Peano's
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
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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 _______________________
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 _______________________
Isn't the valency of a multi-point limited by the valency of vertices in the unit cell tiling? Square unit cells seem to imply that quadruple points have the most possible degeneracy. Or am I wrong? Considering only tiling topology, it would also be possible to have triple points, but it looks like the symmetry of the unit pattern forbids this. The curve snakes across the unit cell between opposite corners, so divides any Z^2/3^n lattice into subsets 2*Z^2/3^n and 2*Z^2/3^n + 1. One associates to quadruple points and the other to double points. This is already visible in the nice "tetraskelion" pic.
From the perspective of multipoints, Wunderlich "Figur 5" is the most interesting. It looks to have double, triple, and quadruple points.
--Brad On Tue, Sep 24, 2019 at 7:38 AM Bill Gosper <billgosper@gmail.com> wrote:
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
Also in Bill/Julian's calculation, you can see the double, triple, and quadruple points on the "peanobert" Z-function with steps 1/72. So this appears consistent with Wunderlich F.5, nice! As for nomenclature, the argument against the name "peanobert" is that Hilbert's curve has linear inflation factor 2, incommensurate with 3. If a "peanobert" curve were to exist, a more likely setting would be a replacement grid of one square to 6-by-6. The argument for the name "peanobert"? I don't know. --Brad On Tue, Sep 24, 2019 at 10:24 AM Brad Klee <bradklee@gmail.com> wrote:
Isn't the valency of a multi-point limited by the valency of vertices in the unit cell tiling? Square unit cells seem to imply that quadruple points have the most possible degeneracy. Or am I wrong?
Considering only tiling topology, it would also be possible to have triple points, but it looks like the symmetry of the unit pattern forbids this. The curve snakes across the unit cell between opposite corners, so divides any Z^2/3^n lattice into subsets 2*Z^2/3^n and 2*Z^2/3^n + 1. One associates to quadruple points and the other to double points. This is already visible in the nice "tetraskelion" pic.
From the perspective of multipoints, Wunderlich "Figur 5" is the most interesting. It looks to have double, triple, and quadruple points.
--Brad
On Tue, Sep 24, 2019 at 7:38 AM Bill Gosper <billgosper@gmail.com> wrote:
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
Here <https://www.wolframcloud.com/obj/230e3446-10e1-4e62-a095-edfbe4e07169> are a few more. I see illusions in the last two. The skinny triangles look gray, and gray shadows lurk in the white areas of the last one. I am no longer censoring accidental swastikas. For posterity, it is time to reclaim our ancient fylfot from the Nazis. Why do we no longer shudder at the sight of red disks? —rwg On Tue, Sep 24, 2019 at 5:37 AM Bill Gosper <billgosper@gmail.com> wrote:
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 _______________________
Hi Bill, The pictures are nice and coooool, no need to suppress them. You should just claim that you are competing with the PAUL KLEE zeitgeist, on the topic of developing the most rebellious possible "Pedagogical Sketchbook", see for example: http://youngfatandlazy.blogspot.com/2010/02/swastika-symbolism.html Also, it is well-known that the last great generation of pre-war Italian scientists--including Peano, Castelnuovo, Volterra, Levi-Civita, etc.--were distinctly anti-fascist; though, some blame the Romans for Fascism, and allege that the idea owes back to the years before Anno Domini started. A google search for "Peano and Fascism" returned a couple of interesting-looking hits: https://www5.in.tum.de/~huckle/fasc.pdf https://jcom.sissa.it/sites/default/files/documents/jcom0101%282002%29A03.pd... But I did not have time to look very closely, busy today. --Brad On Thu, Sep 26, 2019 at 1:34 PM Bill Gosper <billgosper@gmail.com> wrote:
Here <https://www.wolframcloud.com/obj/230e3446-10e1-4e62-a095-edfbe4e07169> are a few more. I see illusions in the last two. The skinny triangles look gray, and gray shadows lurk in the white areas of the last one. I am no longer censoring accidental swastikas. For posterity, it is time to reclaim our ancient fylfot from the Nazis. Why do we no longer shudder at the sight of red disks? —rwg
On Tue, Sep 24, 2019 at 5:37 AM Bill Gosper <billgosper@gmail.com> wrote:
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 _______________________
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
-
Bill Gosper -
Brad Klee