Re: [math-fun] "starshot"
This sounds almost identical to the late Robert Forward's Starwisp proposal, except that that would have used microwaves rather than laser light for power and propulsion. Wikipedia is a good starting point for learning about Starwisp.
Suppose we have a rigid surface X of uniform density in R^3, that can be any shape such that it is topologically a 2-disk, part of whose boundary is a curve C lying in a unique plane that does not intersect X anywhere else, and such that when this curve C is placed on a horizontal plane R^2, the shape stands up on its own, stably. What is the mathematical condition that X stand stably when C is placed on R^2 (the xy-plane), it stands stably on its own ? Assuming a vertical gravity. (With "stably" meaning that if X is tilted in any direction on R^2 by some small enough eps > 0, it will return to a standing position with C on R^2 again.) —Dan
I don't have an answer to this, but a misinterpretation of Dan's question leads me to ask, How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop? Can you do better if you are allowed to break the paperclip into finitely many pieces? (The question is assuredly imprecise, but I'm sure Dan and others will figure out the mathematically sweetest interpretation.) Jim Propp On Sun, Apr 17, 2016 at 10:25 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Suppose we have a rigid surface X of uniform density in R^3, that can be any shape such that it is topologically a 2-disk, part of whose boundary is a curve C lying in a unique plane that does not intersect X anywhere else, and such that when this curve C is placed on a horizontal plane R^2, the shape stands up on its own, stably.
What is the mathematical condition that X stand stably when C is placed on R^2 (the xy-plane), it stands stably on its own ? Assuming a vertical gravity.
(With "stably" meaning that if X is tilted in any direction on R^2 by some small enough eps > 0, it will return to a standing position with C on R^2 again.)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
in an ideal situation, a paperclip of total length L can be bent into a structure of height L-epsilon for any epsilon>0. this is achieved by bending three small legs supporting it, each with length somewhere close to epsilon/5. or have i misunderstood your question? erich
How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop?
You've understood my question completely. I just lacked the visual imagination to see how easy the question was. Jim On Mon, Apr 18, 2016 at 1:51 PM, Erich Friedman <erichfriedman68@gmail.com> wrote:
in an ideal situation, a paperclip of total length L can be bent into a structure of height L-epsilon for any epsilon>0. this is achieved by bending three small legs supporting it, each with length somewhere close to epsilon/5. or have i misunderstood your question?
erich
How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop?
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Nice question, Jim. I'm pretty sure the answer is that L/2 - eps is attainable for any eps > 0, but not for eps = 0. (L = clip length.) —Dan
On Apr 18, 2016, at 10:44 AM, James Propp <jamespropp@gmail.com> wrote:
I don't have an answer to this, but a misinterpretation of Dan's question leads me to ask, How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop? Can you do better if you are allowed to break the paperclip into finitely many pieces?
(The question is assuredly imprecise, but I'm sure Dan and others will figure out the mathematically sweetest interpretation.)
On Sun, Apr 17, 2016 at 10:25 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Suppose we have a rigid surface X of uniform density in R^3, that can be any shape such that it is topologically a 2-disk, part of whose boundary is a curve C lying in a unique plane that does not intersect X anywhere else, and such that when this curve C is placed on a horizontal plane R^2, the shape stands up on its own, stably.
What is the mathematical condition that X stand stably when C is placed on R^2 (the xy-plane), it stands stably on its own ? Assuming a vertical gravity.
(With "stably" meaning that if X is tilted in any direction on R^2 by some small enough eps > 0, it will return to a standing position with C on R^2 again.)
Erich has convinced me that one can attain L - eps for any eps > 0. Do you agree? Jim On Mon, Apr 18, 2016 at 2:00 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Nice question, Jim.
I'm pretty sure the answer is that L/2 - eps is attainable for any eps > 0, but not for eps = 0. (L = clip length.)
—Dan
On Apr 18, 2016, at 10:44 AM, James Propp <jamespropp@gmail.com> wrote:
I don't have an answer to this, but a misinterpretation of Dan's question leads me to ask, How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop? Can you do better if you are allowed to break the paperclip into finitely many pieces?
(The question is assuredly imprecise, but I'm sure Dan and others will figure out the mathematically sweetest interpretation.)
On Sun, Apr 17, 2016 at 10:25 AM, Dan Asimov <dasimov@earthlink.net>
wrote:
Suppose we have a rigid surface X of uniform density in R^3, that can be any shape such that it is topologically a 2-disk, part of whose boundary is a curve C lying in a unique plane that does not intersect X anywhere else, and such that when this curve C is placed on a horizontal plane
R^2,
the shape stands up on its own, stably.
What is the mathematical condition that X stand stably when C is placed on R^2 (the xy-plane), it stands stably on its own ? Assuming a vertical gravity.
(With "stably" meaning that if X is tilted in any direction on R^2 by some small enough eps > 0, it will return to a standing position with C on R^2 again.)
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Yes, I see that I overlooked a much better strategy! (So much for being "pretty sure".) —Dan
On Apr 18, 2016, at 11:09 AM, James Propp <jamespropp@gmail.com> wrote:
Erich has convinced me that one can attain L - eps for any eps > 0. Do you agree?
On Mon, Apr 18, 2016 at 2:00 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Nice question, Jim.
I'm pretty sure the answer is that L/2 - eps is attainable for any eps > 0, but not for eps = 0. (L = clip length.)
On Apr 18, 2016, at 10:44 AM, James Propp <jamespropp@gmail.com> wrote:
. . . How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop? Can you do better if you are allowed to break the paperclip into finitely many pieces?
(The question is assuredly imprecise, but I'm sure Dan and others will figure out the mathematically sweetest interpretation.)
make two (tiny) cuts in the bottom of the paperclip and splay 4 tiny legs apart 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 Mon, Apr 18, 2016 at 2:56 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes, I see that I overlooked a much better strategy!
(So much for being "pretty sure".)
—Dan
On Apr 18, 2016, at 11:09 AM, James Propp <jamespropp@gmail.com> wrote:
Erich has convinced me that one can attain L - eps for any eps > 0. Do you agree?
On Mon, Apr 18, 2016 at 2:00 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Nice question, Jim.
I'm pretty sure the answer is that L/2 - eps is attainable for any eps 0, but not for eps = 0. (L = clip length.)
On Apr 18, 2016, at 10:44 AM, James Propp <jamespropp@gmail.com> wrote:
. . . How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop? Can you do better if you are allowed to break the paperclip into finitely many pieces?
(The question is assuredly imprecise, but I'm sure Dan and others will figure out the mathematically sweetest interpretation.)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
To get a structure of height H, for any H, feed paperclip through extruding machine until it has length H+6cm, then curl last 6 cm into a spiral base so it will stand up. 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 Tue, Apr 19, 2016 at 2:50 AM, Neil Sloane <njasloane@gmail.com> wrote:
make two (tiny) cuts in the bottom of the paperclip and splay 4 tiny legs apart
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 Mon, Apr 18, 2016 at 2:56 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes, I see that I overlooked a much better strategy!
(So much for being "pretty sure".)
—Dan
On Apr 18, 2016, at 11:09 AM, James Propp <jamespropp@gmail.com> wrote:
Erich has convinced me that one can attain L - eps for any eps > 0. Do you agree?
On Mon, Apr 18, 2016 at 2:00 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Nice question, Jim.
I'm pretty sure the answer is that L/2 - eps is attainable for any eps 0, but not for eps = 0. (L = clip length.)
On Apr 18, 2016, at 10:44 AM, James Propp <jamespropp@gmail.com> wrote:
. . . How tall a stable structure can you build using an idealized paperclip that you are allowed to bend but not disconnect, subject to the constraint that it must rest stably on a tabletop? Can you do better if you are allowed to break the paperclip into finitely many pieces?
(The question is assuredly imprecise, but I'm sure Dan and others will figure out the mathematically sweetest interpretation.)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (5)
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Dan Asimov -
Erich Friedman -
James Propp -
Keith F. Lynch -
Neil Sloane