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'Carpenters Ruler' Problem Solved
An unnamed correspondent writes: "Three mathematicians just solved the 'carpenters ruler' problem. The carpenters ruler problem is given a chain of linked rods (a carpenters ruler) in two dimensions, can it always be unwound? As it turns out, it can, check here. You might be saying 'so what', but this has potential applications in anything from protein folding to robotic arm movement. Check here for some animations of the carpenters rule in action."
It just so happens that I have a paper napkin right here at my desk, leftover from lunch. I put it down on my desk: lo and behold, it's a plane.
:)
I then unfolded the paper napkin--and keep in mind that unfolding is really just folding in reverse. Lo and behold, it was four layers of thin paper atop each other; unfolded, it had a substantially larger perimeter.
So the simple answer: unfold the damn napkin.
(Extra credit will be given to those who figure out a way to increase the perimeter of an already unfolded napkin!)
Yeah. Unfortunately, my clever proof is much too long to fit on the napkin.
-fb Everything not expressly forbidden is now mandatory.
I think the image zooms in and out towards the ends of the animation, making it look like the segment lengths change. I'm pretty sure the lengths remain fixed relative to each other (as this is the sense of the problem.)
The article mentioned the proof was announced last June. Has it just now been verified or announced to the public, or are we late getting the news?
kugano
Now if they can only do something about carpenter's crack (also applicable to plumbers).
I swear, I've gone through so many quarters that way.
One of the many things I hate. thingsihate.org
Perhaps this could be applied to the traveling salesman problem, (TSP). The solution to that problem would be a crinkled polygon that we now know could be blown-up like a baloon into a convex polygon. Suppose we start with the points in the TSP surrounded by a convex polygon consisting of segments of the n-1 shortest distances among the points. We now attempt to run the Carpenter's ruler procedure in reverse, sucking in the polygon and trying to fit the joints to the points as we go. I think we would have to slide the polgon around some. If we seem stuck we may have to replace a link with a longer link from the distances greater than the shortest n-1. This is, of course very vague, but I wanted to get it posted in case somebody could see some merit in it.
"Obtuse Anger is that which is greater than Right Anger" - Lewis Carroll
I was reading this article and I was saying "hey, this sounds like Erik's work". And it was. Neat.
:-)
Erik is also very intelligent, and has a professional reputation considerbly higher than most 19 year olds I know
Here's Erik's homepage
Actually, this conjecture was proven this summer past. How do I know this? Because we studied it for some time in September. It is an interesting breakthrough, but I seem to recall the conjecture being taken for granted, and thus although more faith can be put into the applications that fall from Carpenter's Rule, nothing new has really been generated, other than proof. Am I being cynical? Yes. We have no proof of gravity's existence in 2 hours, but we take it for granted. Such as it was with Carpenter's Rule (well, maybe hindsight is 20:20, but this is the impression I had of it ...)
Unlike a two-dimensional chain, this knotted, three-dimensional "knitting needle" chain in space can't be untangled.
;-)
Really? So how was it tangled in the first place, then . . .
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Here's a vaguely-related problem that is fun to contemplate over dinner. You've got a square napkin. You can fold it however you want provided that the resulting shape lies in a plane. For example, you could fold a corner over, which reduces the perimeter, and then fold a "sub-corner" back (as students do with homeworks when they don't have a stapler. :-) ), which increases the perimeter again. The question is, can the perimeter of the shape you form by folding ever exceed the perimeter of the original napkin?
The answer turns out to be... Oh! Look at the time! Gotta get going! :-)