Slashdot Mirror
Major Advances In Knot Theory
An anonymous reader sends us to Science News, which is running a survey of recent strides in finding an answer to the age-old question: How many ways are there to tie your shoelaces? "Mathematicians have been puzzling over that question for a century or two, and the main thing they've discovered is that the question is really, really hard. In the last decade, though, they've developed some powerful new tools inspired by physics that have pried a few answers from the universe's clutches. Even more exciting is that the new tools seem to be the tip of a much larger theory that mathematicians are just beginning to uncover. That larger mathematical theory, if it exists, may help crack some of the hardest mathematical questions there are, questions about the mathematical structure of the three- and four-dimensional space where we live. ... Revealing the full ... superstructure may be the work of a generation."
but I'd hardly call it an age old question. Never heard of it.
How many ways are there to tie your shoelaces? The answer is very easy ... knot.
If enithin kan gow rong it whil. (Murfey)
42
Knot many.
... as the world goes down the toilet. Let us ponder how many ways there are to tie our shoes. When we're done with that, we can contemplate how many angels can dance on the head of a pin. Indeed, after that, we can explore even weightier questions. E.g., does a bear shit in the woods? What is the sound of one hand clapping? If a tree falls... oh never mind.
Revealing the full... superstructure may be the work of a generation.
..assuming computers cease making any new advances.
Mathematicians do rely on their ability to spot patterns and sense implications that no computer can likely sift for today. But this will not always be the case.
-- thinkyhead software and media
Loop and Swoop
Bunny Ears
Where's my Nobel
So, can we abbreviate this "knot theory" to "!theory"?
Anybody want my mod points?
When I read things such as this I like to take a moment to let the dumbfounded feeling wash over me.
This is just not that important.
That larger mathematical theory, if it exists, may help crack some of the hardest mathematical questions there are, questions about the mathematical structure of the three- and four-dimensional space where we live...
And how many mathematicians do you know? The moment this question is even close to being answered, one of them will go off and figure out an even harder question to answer.
"Well now they we have the answer in 4 dimension spaces, we will not rest until we can do it in any prime dimensional space."
I kid because I love.
Could someone please point to relevant information on the topic for why this matters? The article seems to be a a lovely history of a couple of polynomials, but does anything here warrant
The payoff from such work may be profound.
to non-specialists?
Prior to developing his fully inflated knot, the Dog will exhibit various signs of extreme, sexual, excitement. He will pace. Lick. Whimper. Mount. Thrust. At any time during such processes as these, the Dog may Ejaculate clear fluid. Is he " Cumming "? No. He is Ejaculating. His body is pumping fluids, said to be natural lubricants, from his penis. This action appears to have no visable bearing on his Mental / physical / nervous / spiritual disposition at all. He's not particularly enjoying this addendum to the proceedings. He's probably not aware it's happening.
After about five seconds of fucking, you may notice the Knot making itself evident. If you are somehow able to manage to whitness this with a reasonably clear head, you may stop to be amazed at how rapidly everything expands? THIS is the Dogs Orgasm taking place, in manifest form. The physical sign that his mind is blowing. So, for certain, will his dick now be.
... and relies on the cunning use of a rabbit, tree, and hole to tie shoelaces.
It must have been something you assimilated. . . .
the inventor of the shoe lace could be the answer to all our four dimensional space quetions?
....untie the knot my cat did with the mop?
This just in: Physicists have just now revealed that String Theory has nothing to do with the fabric of our universe, and everything to do with teaching toddlers how to tie their shoes.
but it will never be able to explain why anyone would want to "tie the knot."
Man, I haven't posted in years... but there's a great book by this title written by two mathematicians. They talk about the topology of knots as well as the history of ties. Which actors/celebrities wore what tie knots, etc.
I can't seem to locate my copy at the moment, but from what I recall, there are an infinite number of potential knots, but they are classified by the number of sequences in them. And within a certain number of steps, (I think 5) there are 85 possible ways to tie a tie. Then they rank them by symmetry and a copule other criteria.
I recommend it to anybody who is interested in this subject. It's out of print, but it's still possible to find a copy for sale online.
I'm just wondering. One never knows with math.
The answer is obvious: there are infinitely many ways to tie shoelaces. I'm pretty sure I can prove this by mathematical induction in no more than half a dozen steps.
Unless you limit the length of the laces, of course, but then the limit will depend on the physical properties of the laces (not just length but also thickness and so forth), at which point the problem loses its mathematical purity and becomes more of a physics problem.
Cut that out, or I will ship you to Norilsk in a box.
In other news, an algorithm was discovered to determine how many angels can dance on the head of a pin.
I prefer the "velcro" theory.
It's all history, man. -anon
Im more worried about the knots that can be tied but not untied. My shoes are about to get the Alexander's universal knot solution.
Just open a draw containing various cables that has been left for a few months - none of them knotted when you put them in but you can bet when you take them out they'll be more knotted than a knotty thing
--- Users are like bacteria -> Each one causing a thousand tiny crises until the host finally gives up and dies.
Noises out of the of Jordan Huubard AND THE STRIKING
In 1988, physicist Edward Witten of the Institute for Advanced Study in Princeton turned Jones' single invariant into a whole zoo of new invariants using a link he found between Jones' method and quantum theory.
Does this mean that Schrodinger's cat now wears a tie with a very, very complicated knot? This could be a revolution in the mathmaticians annual fashion contest!
Any life is made up of a single moment, the moment in which a man finds out, once and for all, who he is.
Obama - I see him as an overhand knot guy.
McCain - a velcro guy for sure.
... if that's your best, your best won't do... - Twisted Sister
I can honestly say that I find this is the most negative piece of software in the K Desktop Environment.
What are the implications for hyperbondage?
She's discovering the joys of shoelaces now, and you want to talk about knots.. Boy, oh boy. She's gonna be a mathematician for sure!
Karma: Chameleon - mostly influenced by bad '80s New Wave music
A Ph.D. will now be required to join the Boy Scouts.
...abbreviate cutting-and-pasting the first paragraph of the article as the Slashdot article to !summary.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
I have a simple proof of such a knot, but the margin contains too few shoelaces to contain it.
PS: When asked to pull yourself up by your bootlaces, you can now ask for the Jones Polynomial required to do this.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
http://www.sierratradingpost.com/p/,83598_Caterpillar-Groundwork-Wellington-Work-Boots-Steel-Toed-For-Men.html?cm_mmc=PaidPlacement-_-Shopzilla-_-WBIZ8-_-Caterpillar_Groundwork_Wellington_Work_Boots_-_Steel-Toed_-_For_ MY WAY!
"Windows is like the faint smell of piss in a subway: it's there, and there's nothing you can do about it." - Charlie Br
Why is this modded "funny"? This is a serious question. Does anyone actually know the answer?
But I digress. If some mathematician can come over with a theory, and sort this mess of knots out, I'm buying the beer.
And pizza
Schroedinger's Brexit: The UK is both in and out of the EU at the same time!
Who's going to join the BOINC project?
I thought this was the sequel to String Theory.
Man I hope that was a typo. Is it possible you've actually gone through life thinking it was called a "draw"?
For those less interested in theory, and more interested in choosing a lacing pattern and a good knot for their shoes, I recommend Ian's Shoelace Site.
uhhh....
the definition of a Knot is something that cannot be tied or untied.
only a tangle can be ties and untied.
Someone with a heavy Massachusetts accent would call it a 'draw'. Similarly, 'Korea' is pronounced like people in the rest of the country would pronounce 'career', and vice-versa. It's a little surreal for a transplant, such as myself.
-Mike
I'm sorry; I don't know what I was thinking!
He's right.
http://en.wikipedia.org/wiki/Knot_(mathematics)
You know, there is a difference between trolling and pointing out the flaws in your reasoning. Just saying.
>>> That larger mathematical theory, if it exists...
Please make sure you never post again on mathematics.
Insofar as a rigorous math proof produced by a human is operating in a formal system (showing that its conclusions logically follow from its premises), it's subject to the same limitations.
Unless you're arguing that there are correct proofs that depend inherently on hand-waving that could not be made more rigorous, which I think few mathematicians would accept.
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
I wouldn't rule it out, as GP said maths is tricky like that. I'm also intruiged if someone can answer it, and would be impressed by the proof required (unless of course the definition of a knot excludes them).
"Mathematicians have been puzzling over" how to tie their shoelaces "for a century or two"??? Think of what humanity could have achieved if they actually spent that amount of time doing something USEFUL. It's just more proof that shoelaces are bullshit!
How about a closed loop with a knot already in it? No way to untie without cutting the loop.
So, I think that your statements are an accurate assessment of things like Computer Algebra Systems. Such systems approach mathematics in a way similar to how humans have traditionally tried to solve mathematics. However, there are other ways of doing mathematics with computers. Such as various systems of simple abstract rules. I'm not saying it will necessarily lead to breakthroughs in traditional areas of mathematics, but, it is one of the few areas of research that is truly trying to approach mathematics in computer-centric way.
I was responding to the claim that it's somehow been "proven" that computers can't do mathematics as well as mathematicians, presumably because of the negative results about formal systems in Goedel's incompleteness theorem. I was pointing out that human mathematicians manipulating formal systems (which is what mathematics consists of) are subject to the same negative results, so this doesn't really prove anything one way or another.
Now whether computers in practice can do interesting mathematics is of course an open problem.
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
"Tracing basic implications" is hardly the only thing computers do in mathematics; there is plenty of work on the "flash of insight" part, which computers have done successfully on a number of occasions. In particular, there's a long body of work on conjecture-generating systems, which don't try to prove things, but look for conjectures that: 1) would be interesting if true; and 2) seem that they could at least plausibly be true. Generating conjectures is historically a large part of the creativity in mathematics, and in some areas, computers are getting good enough at it that professional mathematicians use conjecture-generating software to get ideas for interesting problems to work on or useful lemmas to prove on the way to another problem.
This survey provides a useful overview of some of the work.
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
A rather old problem. The curious thing is that knots are formally described as closed loops, but Alexander 'solved' the problem, by "cutting the Gordian knot".
What's in a sig?
Slashdot having some issue's as well..
1) Tying your shoelaces (but of course no one cares)
2) Studying supercoiling of DNA (how it wraps itself up into a small space yet still wriggles enough to present all of it's length at short notice for interactions with cells' other mechanisms)
3) The geometry of three dimensional space (all closed oriented three dimensional spaces can be constructed from knots and the three dimensional sphere! So knot theory has major applications to 3D geometry)
4) The geometry of four dimensional space (for example, surfaces in 4D spanning between knots can be used to specify exotic smooth structures. The existence of such shocked the world of geometry in the 80's)
5) TQFT, Mirror Symmetry, Quantum Gravity etc (the tools developed in and around knot theory are one facet of a huge push in mathematics to forge a better understanding of some of the deepest ideas in modern theoretical physics)
It's not all just "brain-wanking".
I'm no mathematician, but I think that all knots (in the traditional definition, ie one string with two ends) can be undone in theory by taking one end and using the string as a path, feeding the end back along its own route; in practice, friction is a problem~
I mod down anyone who says "I will be modded down for this", regardless of the rest of their comment
I thought there were just two: separately, and together.
You might pronounce it that way, but you certainly wouldn't write it.
"limited only by string length". Exactly. "If you can form coded sequences, you can code both random numbers and irrational numbers". Well, only sort of - what you'd have to do is come up with some sort of coding that allows you to *reference* values - so for instance, being able to encode a square-root sign followed by a 2, or for getting pi/4, or so on.
Because you're never gonna get a string long enough to actually spell out sqrt(2) or pi directly.
Of course, once you go that route, some fool is going to Godelize it and hand you a shoestring that says "This number can't be written out on a shoestring". ;)