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Major Advances In Knot Theory
Posted by
kdawson
on Sat Nov 01, 2008 02:31 PM
from the if-it's-not-theory-then-it-must-be-practice dept.
from the if-it's-not-theory-then-it-must-be-practice dept.
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."
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Submission: Unknotting Knot Theory by Anonymous Coward
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An easy answer (Score:5, Funny)
How many ways are there to tie your shoelaces? (Score:3, Insightful)
42
Re:How many ways are there to tie your shoelaces? (Score:5, Funny)
Parent
Re: (Score:3)
Wouldn't it be awesome, though, if the answer really was 42.
Re: (Score:3, Interesting)
Limited only by string length,
since you can alternate slip knots with square knots, you can form coded sequences.
If you can form coded sequences, you can code both random numbers and irrational numbers.
If you can code irrational numbers, like Ummm Hey whats that double T Symbol at the icon for the story
Jackson Pollock would be proud. ( Some call it pleasing, I call it vomit )
Unless... (Score:5, Insightful)
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.
Re: (Score:3, Funny)
Yes, if we discover hard AI and experience a singularity then mathematicians will be obsolete. Of course, so will the rest of us. I'm still going in to work on Monday. How about you?
Re:Unless... (Score:5, Insightful)
No. As a professional computer scientist, I think it is safe to say mathematicians are about the last people in the world to be in danger of losing their job to computers.
If there's one thing computer science algorithmic theory has told us, it's that computers absolutely do have a limit on what they can do, no matter how fast the microchip gets. Complete searches (and that is what we're talking for computer proofs) are NOT getting any more feasible over time. 2^10000 branches will never be traversable.
Pretty much the best possible scenario for computer proofs is basic geometry. After all, in US high school, students are taught "2-column" proofs that a computer could actually handle. And even here, computers suck compared to mediocre mathematicians. Why? Because anybody can trace basic implications like a computer does - that's the easy part. The ONLY real hard part is the flash of insight that computers can never do - e.g. why don't we consider this point that is only tangentially related and see how it somehow holds all the structure to solving the problem.
Once you get into modern math, say knot theory, computers are completely hosed. A math paper might be 100 pages of prose, 80% of which might be insights like that thing above, and 20% of which might be basic implications that a computer can handle. And actually, it couldn't, because 20 pages in prose = 2000 pages in logic statements, and a computer will never be able to traverse that deep.
There's a reason that every important computer proof up until now has relied on 0 insight from the computer... even something like the 4-color theorem is only using a computer to algorithmically check a finite number of trivial cases that would be impractical to check by hand. This approach does not generalize to making mathematicians obsolete.
Parent
not the only thing computers do (Score:4, Interesting)
"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 [vcu.edu] provides a useful overview of some of the work.
Parent
Re:Unless... (Score:4, Funny)
But, mathematicians have already proved that a computer will never be able to take a mathematician's job.
Parent
Re: (Score:3, Informative)
Godel doesn't say that an infinite number of propositions cannot be proved from a finite number of axioms. An infinite number of propositions about geometry can be proven from the handful of axio
!theory (Score:5, Funny)
Solution already patented in 1996 ... (Score:5, Funny)
does this mean? (Score:3, Funny)
Re: (Score:3, Funny)
Yes, String Theory research will be replaced by Tangled Shoelace Theory - the theory that the space-time continuum is in fact a giant cosmic tangle of shoelaces, and that these shoelaces only get untangled in the presence of a large gravitational object, thus causing space-time curvature. In the presence of a massively strong gravitational object such as a black hole, these shoelaces actually break in half, with one half going into the black hole and the other half left dangling in this universe. Thus we s
Ok Great, but can this be used to..... (Score:3, Funny)
....untie the knot my cat did with the mop?
Clandestine Shoelaces (Score:3, Funny)
The 85 Ways to Tie a Tie (Score:5, Informative)
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.
Re:The 85 Ways to Tie a Tie (Score:4, Informative)
Linky. [abebooks.com]
Parent
Can There Be a Knot that Cannot Be Tied or Untied? (Score:5, Funny)
I'm just wondering. One never knows with math.
Re: (Score:3, Funny)
Mod Parent Up (or me!) (Score:3, Informative)
He's right.
http://en.wikipedia.org/wiki/Knot_(mathematics) [wikipedia.org]
wrong theory (Score:5, Funny)
Just looking down at the floor ... (Score:3, Funny)
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
Practical shoelace advice (Score:5, Informative)
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. [fieggen.com]
A few applications of knot theory (Score:5, Informative)
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".
Re:That may be interesting to knot theorists (Score:5, Funny)
e can't be serious.
Parent
Re:That may be interesting to knot theorists (Score:5, Funny)
> e can't be serious.
of course knot. e can't even round correctly. should be 2.7183. damn truncator.
Parent
Re: (Score:3, Informative)
Re:That may be interesting to knot theorists (Score:5, Funny)
but I'd hardly call it an age old question. Never heard of it.
Does that mean you're knot interested in it?
Parent
Re: (Score:3, Interesting)
You sound like you'd take the single most important^h^h^h^h^h^h^h^h^h publicized problem of the day and have everyone working on it, ignoring all of the other interesting stuff that might be possible.
Yes, there are weighty problems in the world, and I'm not trying to dismiss them. Thinking about them exclusively, however, will recover the now but it won't provide any advancement for the future.
Let's do both.
Re:This is so very important... (Score:5, Funny)
Let me introduce you to ^W.
It's a great tool for those writing pseudo-ironic posts who are, at the same time, concerned with the preservation of the valuable resource of ones and zeroes...
Parent
Re:This is so very important... (Score:5, Insightful)
The world has been in far worse situations than it's in now. The transient problems of immediate political and social realities shouldn't stop a few people from investigating nature's deep questions via science and mathematics.
Parent
Re:This is so very important... (Score:5, Insightful)
Why? You made a whiny, irrelevent complaint that dismisses the role of pure research in the larger advancement of our knowledge of how the universe works... the very sort of thing that always plays a role in advancing our ability to make more efficient use of energy, more realistic predictions about the behavior of complex systems, and more innovative technological use of things we think we have already fully, or most effectly exploited. This whole "the human race is incapable of doing two things at once" BS never ceases to amaze me. How do you even get out of bed in the morning? Make coffee... take a crap... which to do first? Gaah! I'm paralyzed! Which is the most important fish to fry?
In other words, you're scare mongering and - if we can assume you have a passable IQ which would suggest you might know better - clearly trolling. And, voila, you were thusly modded.
Parent
Re: (Score:3, Funny)
This whole "the human race is incapable of doing two things at once" BS never ceases to amaze me. How do you even get out of bed in the morning? Make coffee... take a crap... which to do first? Gaah! I'm paralyzed! Which is the most important fish to fry?
Er... are you saying there's a way to take a crap and make coffee at the same time? I'm curious, but at the same time I don't think I want to know...
Re:This is so very important... (Score:5, Interesting)
A mere comment about priorities, relative importance of issues, and so forth. In any case, I was not the only one to make such a comment.
Frankly, mathematics is more important than any other issue. You just fail to realize the practical applications that mathematics has in everyone's life. The most basic reason that anyone on earth has a standard of living above that of hunter gatherers is because of mathematics; knowing seasons and how to plant crops relied on rudimentary mathematics, and modern farming relies on advanced chemistry and biology, which have as their basis the mathematics of stoichiometry and statistics. Not to mention engineering which makes heavy use of mathematics and physics in order to create the machines necessary for our massive population.
In short, I'd rather see advances in mathematics than I would the elimination of world hunger; without further mathematical and scientific discoveries, even nations with plenty will just exhaust their resources and revert to poverty and starvation.
Parent
Re: (Score:3, Funny)
I'm getting too old for this slashdot shit, I guess.
+ 1 insightful
Re: (Score:3, Insightful)
Yes, this is important.
What do you think where new ideas on saving the world or building a better one will come from? TV studios? Politicians? Hollywood?
Research like this is the foundation of all progress. Note: Not this one specifically, I said "like" this. A lot of the things that you probably wouldn't live very well without started out as ideas with no visible use.
Re:This is so very important... (Score:5, Insightful)
Suppose you tell us all how solving this knotty problem will help anyone or anything.
Let's pretend we're in the early 1700s. Leonhard Euler is writing the first ever paper on a field of study called Graph Theory. Simply put, he's figuring out answers to questions about how to arrange circles and lines. Meanwhile, there's fucking WARS going on (Polish succession is going on concurrent to writing this paper; Seven Years' war happens a couple decades later). There are goddamn wars on Euler's front door, and he's writing papers about lines and circles?! What a prick.
Oh, by the way, without Euler's work we wouldn't have computers, organized roads, efficient data models, efficient sorting algorithms, or countless other instruments that are critical to today's society.
Don't trivialize work that you don't understand.
Parent
Re: (Score:3)
OK, but *apart* from computers, organized roads, efficient data models, efficient sorting algorithms, and countless other instruments that are critical to today's society, what has Rome^h^h^h^hresearch ever done for us???
Re: (Score:3, Funny)
wow. mathematicians make such trigger-happy moderators.
modded troll in 3, 2, 1...
Re:The hardest math (Score:5, Insightful)
Parent
Re:The hardest math (Score:5, Insightful)
Hard problems are only hard because we're using the wrong tools.
Parent
Re:The hardest math (Score:4, Funny)
How is that hard? He just has to go through his address book, ask each person what they do and every time one says "mathemetician" he adds 1.
Parent
Re: (Score:3, Funny)
Re: (Score:3, Insightful)
Oh really? Would you also say studying topology in general is unimportant? Why or why not? Since you're able to discern which branches of mathematics aren't "important", you're clearly a mathematical authority, so please feel free to enlighten us.
Re:Things like this... (Score:5, Interesting)
This is just not that important.
Are you sure?
When algebra was invented, did people think that was important? What about geometry or calculus? What about number theory? Would Euler's study of the Seven Bridges of Konigsberg have been important to you? Probably not. But it did lay the foundations for modern graph theory which engineers use to design computer networks.
Parent
Re:Things like this... (Score:5, Interesting)
Back when I was going to school for my Comp Sci degree, I was force-fed a lot of calculus.
Roughly twice as much calculus as was typical, because my disinterest (and the resultant lack of success) required me to take almost every single calculus course twice.
No sooner was I free of school than I brain-dumped every single last integral, deriviative, partial derivative, chain rule, trigometric identity... the lot of it. Good riddance to bad rubbish.
And then, some time later, I was trying to make my race car go faster. The problem was optimising the suspension for maximum grip, and to that end, I had affixed linear potentiometers to my suspension so I could record suspension position during a race.
Pretty soon, I had tons of data relating position to time. Pretty graphs, but aside from max/min/mean deflection data, pretty useless.
Until I started thinking about "position to time... position to time... where had I heard that before?"
That's right - my old arch-nemesis, calculus, suddenly proved useful. Deriving that position information gave me suspension velocity, and suddenly I knew EXACTLY what suspension velocities the car was seeing in actual competition. Given that I had a device that measured shock force as a function of velocity (that's how a shock works) I could now tune shocks independant of the driver's ass-dyno.
That resulted in a HUGE leap forward in my performance.
Don't dis abstract math - you never know when it'll pay off.
DG
Parent
Re:Things like this... (Score:5, Funny)
Back when I was going to school for my Elementary School diploma, I was force-fed a lot of arithmetic.
Roughly twice as much as was typical, because my disinterest (and the resultant lack of success) required me to take almost every grade twice.
No sooner was I free of school than I brain-dumped every single addition, multiplication, subtraction, division, counting... the lot of it. Good riddance to bad rubbish.
And then, some time later, I was trying to make my paycheck go farther. The problem was optimising the spending for maximum personal happiness, and to that end, I had collected all of my receipts so that I could record where I was spending my money during the month.
Pretty soon, I had tons of data indicating where my money was going. Pretty numbers, but aside from a few expensive items, pretty useless.
Until I started thinking about what I could do with a set of numbers.
That's right - my old arch-nemesis, arithmetic, suddenly proved useful. Summing the money spent in different categories gave me totals, and suddenly I knew EXACTLY where my money was going in an actual month. Given that I had measured how much money was spent on each purchase (that's how receipts work) I could now properly budget my spending.
That resulted in a HUGE leap forward in my quality of life.
Don't dis abstract math - you never know when it'll pay off.
AC
Parent
Re:Things like this... (Score:4, Funny)
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.
You only say that because you have yet to be involved in a serious shoe-tying accident.
Parent