[math-fun] Lucky accident
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --rwg
Nice - I like it! I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them: http://www.karzes.com/ghack14-1.gif For more examples, see: http://www.karzes.com/dragon.html They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy. Tom Bill Gosper writes:
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --rwg
Here's a twindragon version of the same thing, formed by gluing together two Heighway dragons: http://www.karzes.com/ghack14-twin-1.gif Given that this is now a closed curve, it would almost certainly look better filled, but, still lazy... Tom Tom Karzes writes:
Nice - I like it!
I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them:
http://www.karzes.com/ghack14-1.gif
For more examples, see:
http://www.karzes.com/dragon.html
They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy.
Tom
Bill Gosper writes:
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --rwg
Here's an inside-out version, showing that the inside and the outside, while visually distinct, may be freely interchanged: http://www.karzes.com/ghack14-inv-twin-1.gif Tom Tom Karzes writes:
Here's a twindragon version of the same thing, formed by gluing together two Heighway dragons:
http://www.karzes.com/ghack14-twin-1.gif
Given that this is now a closed curve, it would almost certainly look better filled, but, still lazy...
Tom
Tom Karzes writes:
Nice - I like it!
I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them:
http://www.karzes.com/ghack14-1.gif
For more examples, see:
http://www.karzes.com/dragon.html
They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy.
Tom
Bill Gosper writes:
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --rwg
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
--
Lovely pics! Do any of those curves give rise to interesting number sequences (giving say the number of square cells, or blobs, or curve segments), as they grow? For some animations in the OEIS, which certainly are associated with sequences, go to https://oeis.org/A139250, scroll down to the second link (David Applegate, The movie version), pick any of a hundred different sequences from the drop-down menus, and click Next repeatedly - or click Run. The E-toothpick sequence, A161328, is especially pretty - and no recurrence is known for it .... Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Wed, Dec 27, 2017 at 8:08 AM, Tom Karzes <karzes@sonic.net> wrote:
Here's an inside-out version, showing that the inside and the outside, while visually distinct, may be freely interchanged:
http://www.karzes.com/ghack14-inv-twin-1.gif
Tom
Tom Karzes writes:
Here's a twindragon version of the same thing, formed by gluing together two Heighway dragons:
http://www.karzes.com/ghack14-twin-1.gif
Given that this is now a closed curve, it would almost certainly look better filled, but, still lazy...
Tom
Tom Karzes writes:
Nice - I like it!
I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them:
http://www.karzes.com/ghack14-1.gif
For more examples, see:
http://www.karzes.com/dragon.html
They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy.
Tom
Bill Gosper writes:
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --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
I would like to mention that the "D-toothpick" form happens to be independently invented by me in the early 1980s as a means of doodling on arithmetic paper during boring high school classes and implemented as a 1995 IOCCC entry: http://ioccc.org/years.html#1995_leo Interesting variations include deciding at each step how the branches should grow: in both directions, to the left, or to the right. This gives rise to a multitude of beautiful patterns, when the sequence is periodic with a short period, as well as when it is pseudo-random. A question to ponder is, what is the minimum percentage of "both" in a random sequence so that the tree is expected to grow infinitely. Leo On Wed, Dec 27, 2017 at 8:42 AM, Neil Sloane <njasloane@gmail.com> wrote:
Lovely pics!
Do any of those curves give rise to interesting number sequences (giving say the number of square cells, or blobs, or curve segments), as they grow?
For some animations in the OEIS, which certainly are associated with sequences, go to https://oeis.org/A139250, scroll down to the second link (David Applegate, The movie version), pick any of a hundred different sequences from the drop-down menus, and click Next repeatedly - or click Run.
The E-toothpick sequence, A161328, is especially pretty - and no recurrence is known for it ....
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Dec 27, 2017 at 8:08 AM, Tom Karzes <karzes@sonic.net> wrote:
Here's an inside-out version, showing that the inside and the outside, while visually distinct, may be freely interchanged:
http://www.karzes.com/ghack14-inv-twin-1.gif
Tom
Tom Karzes writes:
Here's a twindragon version of the same thing, formed by gluing together two Heighway dragons:
http://www.karzes.com/ghack14-twin-1.gif
Given that this is now a closed curve, it would almost certainly look better filled, but, still lazy...
Tom
Tom Karzes writes:
Nice - I like it!
I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them:
http://www.karzes.com/ghack14-1.gif
For more examples, see:
http://www.karzes.com/dragon.html
They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy.
Tom
Bill Gosper writes:
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Leo, Great to hear that! If you could formulate a link to that IOCCC entry I would be delighted to add it to the OEIS! Something like this: Leo Broukhis, URL, Title, Date I know you said the URL was : http://ioccc.org/years.html#1995_leo but I got lost there! It looks like there are at least two versions of the D-toothpick sequence in the OEIS, namely https://oeis.org/A194270 and A194700. (You can run them all from one of the drop-down menus in A139250.) I guess yours was the first one, A194270. Of course the big question is, how many toothpicks are there after n generations? Did you ever work out a recurrence for any of the variations you considered? (I never studied the D-toothpick structure. A139250 was not too hard to solve, but others eluded us.) You said: "... This gives rise to a multitude of beautiful patterns, when the sequence is periodic with a short period" Which one was that, do you recall? Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Dec 28, 2017 at 2:31 AM, Leo Broukhis <leob@mailcom.com> wrote:
I would like to mention that the "D-toothpick" form happens to be independently invented by me in the early 1980s as a means of doodling on arithmetic paper during boring high school classes and implemented as a 1995 IOCCC entry: http://ioccc.org/years.html# 1995_leo
Interesting variations include deciding at each step how the branches should grow: in both directions, to the left, or to the right. This gives rise to a multitude of beautiful patterns, when the sequence is periodic with a short period, as well as when it is pseudo-random.
A question to ponder is, what is the minimum percentage of "both" in a random sequence so that the tree is expected to grow infinitely.
Leo
On Wed, Dec 27, 2017 at 8:42 AM, Neil Sloane <njasloane@gmail.com> wrote:
Lovely pics!
Do any of those curves give rise to interesting number sequences (giving say the number of square cells, or blobs, or curve segments), as they grow?
For some animations in the OEIS, which certainly are associated with sequences, go to https://oeis.org/A139250, scroll down to the second link (David Applegate, The movie version), pick any of a hundred different sequences from the drop-down menus, and click Next repeatedly - or click Run.
The E-toothpick sequence, A161328, is especially pretty - and no recurrence is known for it ....
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Dec 27, 2017 at 8:08 AM, Tom Karzes <karzes@sonic.net> wrote:
Here's an inside-out version, showing that the inside and the outside, while visually distinct, may be freely interchanged:
http://www.karzes.com/ghack14-inv-twin-1.gif
Tom
Tom Karzes writes:
Here's a twindragon version of the same thing, formed by gluing together two Heighway dragons:
http://www.karzes.com/ghack14-twin-1.gif
Given that this is now a closed curve, it would almost certainly look better filled, but, still lazy...
Tom
Tom Karzes writes:
Nice - I like it!
I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them:
http://www.karzes.com/ghack14-1.gif
For more examples, see:
http://www.karzes.com/dragon.html
They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy.
Tom
Bill Gosper writes:
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --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
_______________________________________________ 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
Hello Neil, http://ioccc.org/1995/leo.hint describes the obfuscated program and the algorithm (in the traditional IOCCC "humorous" style). The actual program is http://ioccc.org/1995/leo.c (it requires http://ioccc.org/1995/Makefile to build). Unfortunately, I've forgotten what most secret options of the program do. I'll try to revive it in the coming weeks. My two versions were "non-hairy", when any intersection of the toothpicks stops the subsequent growth, and "hairy", when only the intersections at integer grid points matter. The "non-hairy" version matches D-toothpicks exactly. I did not think of the toothpick counts then, and for fractional or overlapping toothpicks it is not easy to define. It is my loss that I haven't analyzed it earlier. Having come up with the pattern in a biology class, when [dichotomous branching]( https://socratic.org/questions/what-is-dichotomous-branching) in lower plants was described, I was thinking of it mostly from an aesthetic point of view. For example, alternating "both" and "right" results in tiling of quarters of the plane. After reviving my program, I'll be able to tell more, if nobody beats me to it. Best regards, Leo On Thu, Dec 28, 2017 at 12:15 AM, Neil Sloane <njasloane@gmail.com> wrote:
Leo, Great to hear that!
If you could formulate a link to that IOCCC entry I would be delighted to add it to the OEIS! Something like this:
Leo Broukhis, URL, Title, Date
I know you said the URL was : http://ioccc.org/years.html#1995_leo but I got lost there!
It looks like there are at least two versions of the D-toothpick sequence in the OEIS, namely https://oeis.org/A194270 and A194700. (You can run them all from one of the drop-down menus in A139250.) I guess yours was the first one, A194270.
Of course the big question is, how many toothpicks are there after n generations? Did you ever work out a recurrence for any of the variations you considered? (I never studied the D-toothpick structure. A139250 was not too hard to solve, but others eluded us.) You said: "... This gives rise to a multitude of beautiful patterns, when the sequence is periodic with a short period" Which one was that, do you recall?
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Dec 28, 2017 at 2:31 AM, Leo Broukhis <leob@mailcom.com> wrote:
I would like to mention that the "D-toothpick" form happens to be independently invented by me in the early 1980s as a means of doodling on arithmetic paper during boring high school classes and implemented as a 1995 IOCCC entry: http://ioccc.org/years.html# 1995_leo
Interesting variations include deciding at each step how the branches should grow: in both directions, to the left, or to the right. This gives rise to a multitude of beautiful patterns, when the sequence is periodic with a short period, as well as when it is pseudo-random.
A question to ponder is, what is the minimum percentage of "both" in a random sequence so that the tree is expected to grow infinitely.
Leo
On Wed, Dec 27, 2017 at 8:42 AM, Neil Sloane <njasloane@gmail.com> wrote:
Lovely pics!
Do any of those curves give rise to interesting number sequences (giving say the number of square cells, or blobs, or curve segments), as they grow?
For some animations in the OEIS, which certainly are associated with sequences, go to https://oeis.org/A139250, scroll down to the second link (David Applegate, The movie version), pick any of a hundred different sequences from the drop-down menus, and click Next repeatedly - or click Run.
The E-toothpick sequence, A161328, is especially pretty - and no recurrence is known for it ....
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Dec 27, 2017 at 8:08 AM, Tom Karzes <karzes@sonic.net> wrote:
Here's an inside-out version, showing that the inside and the outside, while visually distinct, may be freely interchanged:
http://www.karzes.com/ghack14-inv-twin-1.gif
Tom
Tom Karzes writes:
Here's a twindragon version of the same thing, formed by gluing together two Heighway dragons:
http://www.karzes.com/ghack14-twin-1.gif
Given that this is now a closed curve, it would almost certainly look better filled, but, still lazy...
Tom
Tom Karzes writes:
Nice - I like it!
I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them:
http://www.karzes.com/ghack14-1.gif
For more examples, see:
http://www.karzes.com/dragon.html
They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy.
Tom
Bill Gosper writes: > Looking to illustrate spacefilling by using continuously varying colors, > with no tricky phases > and sampling frequencies, I found > gosper.org/blackdrag.png > which looks nice all black yet shows the course of the spacefill, using > only actual dyadic rationals from the Dragon > recursion. The trick is to replace every triad of consecutive points by a > triangle with those vertices. It might be > tweakable into one of those inside|outside > https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg > Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png > isn't confusing enough. --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
_______________________________________________ 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I've cleaned up my Heighway/twin dragon curve page a bit, and added filled twindragons. Here are some examples: http://www.karzes.com/dragon/dragon.html?style=rect&form=f http://www.karzes.com/dragon/dragon.html?style=dart&form=f http://www.karzes.com/dragon/dragon.html?style=rect&variant=1&form=h They're all the same page, with parameters that you can vary. Tom Tom Karzes writes:
Here's an inside-out version, showing that the inside and the outside, while visually distinct, may be freely interchanged:
http://www.karzes.com/ghack14-inv-twin-1.gif
Tom
Tom Karzes writes:
Here's a twindragon version of the same thing, formed by gluing together two Heighway dragons:
http://www.karzes.com/ghack14-twin-1.gif
Given that this is now a closed curve, it would almost certainly look better filled, but, still lazy...
Tom
Tom Karzes writes:
Nice - I like it!
I have some Heighway Dragon renderings that use different styles for left- vs. right-turns, resulting in a nice inside vs. outside distinction. Here's one of them:
http://www.karzes.com/ghack14-1.gif
For more examples, see:
http://www.karzes.com/dragon.html
They would probably be more dramatic if I shaded one side of the curve, but I'm too lazy.
Tom
Bill Gosper writes:
Looking to illustrate spacefilling by using continuously varying colors, with no tricky phases and sampling frequencies, I found gosper.org/blackdrag.png which looks nice all black yet shows the course of the spacefill, using only actual dyadic rationals from the Dragon recursion. The trick is to replace every triad of consecutive points by a triangle with those vertices. It might be tweakable into one of those inside|outside https://pictures.abebooks.com/isbn/9780262630221-us-300.jpg Minsky-Papert perceptron confusers. Not that gosper.org/mediandrag2.png isn't confusing enough. --rwg
Nice! The Heighway dragon is the smallest (nontrivial) element in the set of curves generated by "folding morphisms" (that name is by Michel Dekking). A recent search gave a total of 441272 such curves for all orders R <= 53, see https://oeis.org/A296148 I can make the data available if you want to have a rather insane amount of possibilities added to your page. Stopping at, say, order R=13 already gives a pretty decent collection of shapes. Best regards, jj * Tom Karzes <karzes@sonic.net> [Feb 06. 2018 17:37]:
I've cleaned up my Heighway/twin dragon curve page a bit, and added filled twindragons. Here are some examples:
http://www.karzes.com/dragon/dragon.html?style=rect&form=f http://www.karzes.com/dragon/dragon.html?style=dart&form=f http://www.karzes.com/dragon/dragon.html?style=rect&variant=1&form=h
They're all the same page, with parameters that you can vary.
Tom [...]
participants (5)
-
Bill Gosper -
Joerg Arndt -
Leo Broukhis -
Neil Sloane -
Tom Karzes