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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."
Haven't you ever read Dirk Gently? Remember the bit about the sofa in the stairwell...
Connelly is my Math professor for my Linear Algebra class this semester. Pretty cool to see his name pop up on /. :)
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-Jeff
-Vercingetorix
-Vercingetorix
"Necessitas non habet legem." -St. Augustine
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!)
Gold does not have intrinsic value. Like everything else, it has value based on supply and demand.
You seem to be confused. The normal usage of "intrinsic value" in economics is that people find it valuable even in the absence of laws/regulations/common consent/consensual hallucinations. Gold DOES have intrinsic value (it's pretty, resistant to corrosion, conducts heat very very well, etc.), as opposed, say, to paper US dollars which do not.
The fact that the price of gold depends on the supply and demand has nothing to do with it. The price of everything depends on supply and demand.
And then what happens when, say, Russia floods the world with cheap gold from their huge reserves?
Russia does not have huge reserves and its gold isn't cheap (meaning the costs of production are quite comparable with the West/Australia/South Africa).
Besides, WHAT would happen? The worst is that we'll have a bit of inflation, not much at all compared to paper dollars inflation that we had in the 70s and the 80s. The supply of gold in the world is quite limited, as opposed to the capacity of the government printing presses.
Note that I am not arguing for the return to a gold standard -- this is an idea the time of which has passed long, long time ago. I just want to point out that if you want to argue about a subject, it helps to have some clue about it.
Kaa
Kaa
Kaa's Law: In any sufficiently large group of people most are idiots.
I believe this was a constructive proof; they not only proved it was possible, but also how to solve for motions. The problem's open status suggests that no such solution was known earlier.
perfect convexification of a polygon would produce a reasonable estimation of a circle. A circle is the maximal area with a given length of edge.
A fractal surface is a way of generating an infinite edge length with a finite area. If you were to convexify such a fractalized area, you would end up with a potentially infinite circumference circle generated from a fractal of area 1 (or any other number you might want to choose.).
For a simple counter-example, consider a star. Convexified to a 'circular' polygon, it would be a roughly circular polygon capable of containing the original star. q.e.d.
`ø,,ø`ø,,ø!
Free Software: Like love, it grows best when given away.
...this is only two dimensions, and a 2D proof doesn't say much about what the case is in 3D. Applications like protein folding and robotic arm movement take place in the real world, which is unfortunately 3D.
Yeah. Unfortunately, my clever proof is much too long to fit on the napkin.
-fb Everything not expressly forbidden is now mandatory.
How about with something based one REAL currency with intrinsic value, like silver and gold,
And then what happens when, say, Russia floods the world with cheap gold from their huge reserves? [which was a very real possibility that was floated during the Y2K thing, which was why smart people did NOT put their money into gold]
Gold does not have intrinsic value. Like everything else, it has value based on supply and demand. Dollars have value because of common consent. The difference is that we, as a country, control the supply and aren't at the mercy of a foreign power who might flood the market with gold.
The Federal Reserve may not be a perfect system, but it's way better than basing it on arbitrary metals. Heck, how about basing the currency on oil? Think about how insane that would be, and apply the same thinking to gold or silver.
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Sometimes it's best to just let stupid people be stupid.
-Vercingetorix
-Vercingetorix
"Necessitas non habet legem." -St. Augustine
That's unlikely to happen with Gold, since gold is simply hard to find. Although it's possible to go off of the gold standard, it requires a conscious effort, rather than simple stupidity.
There is, however, at least one counter-example which almost proves the point. History records A gold-standard inflation problem which occurred when Europe found, and plundered, the New World. Europe's supply of gold expanded massively. Those countries which were not in on the plunder suffered from the sudden shift in the supply/demand curve of gold.
The extra supply, however, consisted of thousands of years worth of americas' gold mining. Once the plunder was complete, things settled down again.
`ø,,ø`ø,,ø!
Free Software: Like love, it grows best when given away.
The animation appears to have variable-length rulers... is it supposed to be like that?
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Napster-to-go says "Fill and refill your compatible MP3 player", which is a lie. It's not MP3. It's WMA with DRM.
The lives enjoyed by many math mysteries are being shortened by the technological age. Yet we do not create many new questions to be raised. With such a lopsided affair, what will drive future mathematics?
Pax Digitalia
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
Any geek worth his NaCl should have a subscription to Science News (http://www.sciencenews.org). It's a weekly publication, and covered this a couple of weeks ago.
It also covered a possible loophole in the second law of thermodynamics that might make a perpetual motion machine of the second type possible, using Quantum Dynamics.
Go take a look.
www.eFax.com are spammers
unwind a Keren Carpenter ruler? It is the same problem until the tape stretches thin and becomes brittle.
"Trademarks are the heraldry of the new feudalism."
Dood.. to solve something, you first need to know how to solve it...
:(
You need like a set of steps worked out that when followed, will lead to a solution. It's called an 'algorithm' man! You must first come up with an algorithm, before you can wack it onto a huge Beowulf cluster - and coming up with algorithm is the HARDEST part
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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This message has been ROT-13 encrypted twice for higher security.
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! :-)
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?
Nope. There's only so much of the original parameter, (I'll call this the "sharp" edge) and when you fold the paper over the first time, the "bent" edge is always shorter than the sharp edge you just folded away. Now we'll to try to regain some parameter. When you make the second fold, you gain back a portion of the sharp edge that's larger than the bent edge you are now losing. But look -- you should see a triangle on each side of the folded area, each composed of one line of the bent edge and two of the sharp edges. The "inner" two lines of the triangle are the sharp edge. Still with me? It's a rule that the sum of the length of two sides of a triangle are always longer than the third. So the length of the remaining bent edge on the parameter is smaller than the sharp edge you lost, and the second fold only adds back a fraction of the sharp edge. You can never regain back even the original length. That triangle cost you too much space!
Remember "Bring 'em on"? *sigh