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.)