[math-fun] What area is filled by the Dragon joining 0+0i to 1+0i?
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research". It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither. You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>) The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
The Heighway area should be .5 and the twindragon area should be 1. You can get them directly from the tiling polygons. Heighway: https://www.karzes.com/xfract/img/dragon.html Polygon: (0, 0) (.6, -.2) (1, 0) (.8, .4) (.2, .6) Area: .5 Twin: https://www.karzes.com/xfract/img/twindragon.html Polygon: (.2, -.4) (.8, -.6) (1.2, -.4) (.8, .4) (.2, .6), (-.2, .4) Area: 1 Tom Bill Gosper writes:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
Sweet! Doesn't that mean that the area of all those "dragons", when looked upon as fundamental tile of the corresponding numeration system is = 1 ? Anyway, here is Dekking morphism: a |--> ab A |--> BA b |--> Ab B |--> Ba where I wrote A for a^(-1) and B for b^(-1). In the image https://jjj.de/tmp-math-fun/twindragon-boundary-dekking.pdf I have not done the deletions aA = Aa = bB = Bb = empty word, thus the "excursions" into and out off the twindragon. Best regards, jj Note to myself: zrender -d=4 -m='1 12 2 32 3 43 4 41' -e=0.1 -w=2 -a='1234' -k=-1 -i=5 --arrows --letters where 1 = a, 2 = b, 3 = A, 4 = B * Tom Karzes <karzes@sonic.net> [Nov 09. 2019 16:53]:
The Heighway area should be .5 and the twindragon area should be 1. You can get them directly from the tiling polygons.
Heighway:
https://www.karzes.com/xfract/img/dragon.html
Polygon: (0, 0) (.6, -.2) (1, 0) (.8, .4) (.2, .6) Area: .5
Twin:
https://www.karzes.com/xfract/img/twindragon.html
Polygon: (.2, -.4) (.8, -.6) (1.2, -.4) (.8, .4) (.2, .6), (-.2, .4) Area: 1
Tom
Bill Gosper writes: [...]
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
The link to the Knuth paper doesn't work. (Because of spaces I think) On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com> wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
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On 2019-11-09 09:09, Paul Palmer wrote:
The link to the Knuth paper doesn't work. (Because of spaces I think)
Because of spaces where there should be brain tissue between the ears of the idiots behind xmission.com . It works in my gmail outbox. Tnx for the heads-up. You can recover by reconstructing the url in your browser's url pane. —rwg
On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com> wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
Tried everything I can think of but can't make it work with chrome on my phone (android) Oh well On Sat, Nov 9, 2019, 12:05 PM rwg <rwg@ma.sdf.org> wrote:
On 2019-11-09 09:09, Paul Palmer wrote:
The link to the Knuth paper doesn't work. (Because of spaces I think)
Because of spaces where there should be brain tissue between the ears of the idiots behind xmission.com . It works in my gmail outbox. Tnx for the heads-up. You can recover by reconstructing the url in your browser's url pane. —rwg
On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com> wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon
Curves,
of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
Maybe try a shortened url: https://support.google.com/faqs/answer/190768?hl=en On Sat, Nov 9, 2019 at 18:43 Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Tried everything I can think of but can't make it work with chrome on my phone (android)
Oh well
On Sat, Nov 9, 2019, 12:05 PM rwg <rwg@ma.sdf.org> wrote:
On 2019-11-09 09:09, Paul Palmer wrote:
The link to the Knuth paper doesn't work. (Because of spaces I think)
Because of spaces where there should be brain tissue between the ears of the idiots behind xmission.com . It works in my gmail outbox. Tnx for the heads-up. You can recover by reconstructing the url in your browser's url pane. —rwg
On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com>
wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they
give
the
area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
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http://gosper.org/Number%20representation%20and%20dragon%20curves%20Paperfol... On Sat, Nov 9, 2019 at 4:48 PM Victor Miller <victorsmiller@gmail.com> wrote:
Maybe try a shortened url: https://support.google.com/faqs/answer/190768?hl=en
On Sat, Nov 9, 2019 at 18:43 Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Tried everything I can think of but can't make it work with chrome on my phone (android)
Oh well
On Sat, Nov 9, 2019, 12:05 PM rwg <rwg@ma.sdf.org> wrote:
On 2019-11-09 09:09, Paul Palmer wrote:
The link to the Knuth paper doesn't work. (Because of spaces I think)
Because of spaces where there should be brain tissue between the ears of the idiots behind xmission.com . It works in my gmail outbox. Tnx for the heads-up. You can recover by reconstructing the url in your browser's url pane. —rwg
On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com>
wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they
give
the
area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
That worked. Thank you. I should have thought of that. Sorry to bother everyone. On Sat, Nov 9, 2019, 6:21 PM Mike Stay <metaweta@gmail.com> wrote:
http://gosper.org/Number%20representation%20and%20dragon%20curves%20Paperfol...
On Sat, Nov 9, 2019 at 4:48 PM Victor Miller <victorsmiller@gmail.com> wrote:
Maybe try a shortened url: https://support.google.com/faqs/answer/190768?hl=en
On Sat, Nov 9, 2019 at 18:43 Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Tried everything I can think of but can't make it work with chrome on
my
phone (android)
Oh well
On Sat, Nov 9, 2019, 12:05 PM rwg <rwg@ma.sdf.org> wrote:
On 2019-11-09 09:09, Paul Palmer wrote:
The link to the Knuth paper doesn't work. (Because of spaces I think)
Because of spaces where there should be brain tissue between the ears of the idiots behind xmission.com . It works in my gmail outbox. Tnx for the heads-up. You can recover by reconstructing the url in your browser's url pane. —rwg
On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com>
wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <
representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com
wrote:
> Wikipedia seems not to say. I wonder if expositors of "spacefilling > curves" > really feel in their gut that the space is filled. Or maybe they give the > area, > but Wikipedia censors it as "original research". > > It's probably in Knuth & Davis, Number Representations and Dragon Curves, > of which I have at least 2 copies and can find neither. > > You can guess the answer if you believe the numbers pasted on > Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But > there's a direct approach. > (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>) > > The Dragon's image is dense with triple points and has uncountably many > double points, but I think they have measure zero, and wouldn't affect the > "area" if you counted them thrice and twice. —rwg
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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The link with the spaces worked for me, but I cut it and pasted it to open a new tab. I removed the On Sat, Nov 9, 2019 at 11:09 AM Paul Palmer <paul.allan.palmer@gmail.com> wrote:
The link to the Knuth paper doesn't work. (Because of spaces I think)
On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com> wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Sorry the truncated message. It (the link) worked with the spaces using chrome under windows 10. I copied and pasted (removing the first character) to create a new tab. On Sat, Nov 9, 2019 at 12:45 PM James Buddenhagen <jbuddenh@gmail.com> wrote:
The link with the spaces worked for me, but I cut it and pasted it to open a new tab. I removed the
On Sat, Nov 9, 2019 at 11:09 AM Paul Palmer <paul.allan.palmer@gmail.com> wrote:
The link to the Knuth paper doesn't work. (Because of spaces I think)
On Sat, Nov 9, 2019, 10:58 AM Bill Gosper <billgosper@gmail.com> wrote:
Wow, both Jörg Arndt and (Dekking's student!) Arie Bos sent me photocopies of the Knuth & Davis Dragon paper. (The latter signed by Knuth! <http://gosper.org/Number representation and dragon curves Paperfolding-Knuth.pdf>) But I see no mention that the area is (SPOILER) half the square of the distance between the endpoints. It is amazing how empowering to one's intuition are Julian's Mathematica functions, which permit exploring fractal functions in the continuum. E.g., I also found a pdf colorfully dividing a Dragon's area into equal tenths <http://gosper.org/dekdrag.pdf>. —rwg
On Fri, Nov 8, 2019 at 10:21 PM Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research".
It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither.
You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>)
The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg
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participants (8)
-
Bill Gosper -
James Buddenhagen -
Joerg Arndt -
Mike Stay -
Paul Palmer -
rwg -
Tom Karzes -
Victor Miller