Major Advances In Knot Theory 230
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."
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)
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Wouldn't it be awesome, though, if the answer really was 42.
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how the fuck is 42 insightful? its funny when you read it in that book, but seeing it here again and again is not even funny any more, let alone insightful
Someone's forgotten where their towel is.
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when I first read the main page of slashdot, I said to myself, 42!, It's good to be able to laugh and be silly, and pay homage to the great number 42.
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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 )
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How many combinations of digits does pi have?
One.
3.141592653589793...
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.
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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?
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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
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Computing power has very diminishing returns, you can punch up a possible result in the computer and it'll pound throught it for a few million numbers or run a few million simulations. If it doesn't come back with a counter example, you might be on to something. But the chance that it'll actually give you any more useful information if you could run billions or trillions of tests isn't really all that great. There's the odd case like the four color theorem but it still took a lot more manhours than it took
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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.
Simple Abstract Rules (Score:3, Interesting)
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.
Re:Unless... (Score:4, Funny)
But, mathematicians have already proved that a computer will never be able to take a mathematician's job.
QED (Score:2, Funny)
Loop and Swoop
Bunny Ears
Where's my Nobel
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This one [fieggen.com] is quite good, especially for rock climbing and hiking.
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Why not just use the fieggen shoelace knot [animatedknots.com] itself?
Why wasn't I taught this when I was younger? All these damn posts and not one about a new way to tie your shoes.
!theory (Score:5, Funny)
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I trow knot.
No, but we can... (Score:2)
Solution already patented in 1996 ... (Score:5, Funny)
does this mean? (Score:3, Funny)
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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
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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 see no light as all the shoelaces are now in a tightly tangled ball that has no connection to this universe.
Okay, that's all fine, but... who wears the shoes?
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Turtles :) All the way down :)
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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)
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Every time you tie your shoes, the universe kills a kitten(through 4-dimensional knot strangulation). Think of all the kittens!
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]
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Can There Be a Knot that Cannot Be Tied or Untied? (Score:5, Funny)
I'm just wondering. One never knows with math.
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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
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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
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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.
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uhhh....
the definition of a Knot is something that cannot be tied or untied.
only a tangle can be ties and untied.
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)
Hey, I read XKCD (Score:2)
My 3 year old volunteers to answer the question (Score:2)
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]
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There's even a book on that site about all the ways in which you can tie a shoe!
http://www.fieggen.com/shoelace/iansbook.htm [fieggen.com]
And for those who just want to tie the damn things a bit quicker, there's the "ian knot":
http://www.fieggen.com/shoelace/ianknot.htm [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.
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You wouldn't believe what just thinking about this is doing to my stomach...
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It's making your stomach unsettled?
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Why it's positively tied up in knots!
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.
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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?
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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...
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I wish we would move onto ^X
I'm sick of the ^H and ^W
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.
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.
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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...
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My coffee pot is on a timer so it's already brewing as I take care of other business. next?
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.
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Let politicians work on politics, soldiers work on war, and mathematicians work on math.
I have to say that I think current affairs would at least be more interesting if we had scientists work on politics and politicians work on research.
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I'm getting too old for this slashdot shit, I guess.
+ 1 insightful
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Studying how to tie shoelaces is hardly what 'math guys' do. It's worthy of a slashdot article but I'll wager the article is written by a couple of students who'd had too much beer.
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Being a graduate student in mathematics , I can safely assert that knot theory is actually a significant area of modern mathematics. There are numerous textbooks about it. [google.com]. If you read the article you would know that Jones & Witten received a Fields Medal, which is the most prestigious award in mathematics, for their work on classifying knots.
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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.
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I'm not a neurosurgeon - I'm a computer geek, of sorts, so I program stuff. Does my programming stuff save lives? probably not.. it may make some people's lives easier, but that's about it. So would you tell me to go to medical school and study neurosurgery so I can do something important like save lives?
You're a computer geek and NOT a neurosurgeon? Well then, I'm going to have to cancel my thursday appointment.
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.
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Without x, we would not have y, therefore it's all good.
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wow. mathematicians make such trigger-happy moderators.
modded troll in 3, 2, 1...
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My original post was actually meant to be funny, but... whoosh!
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Don't trivialize work that you don't understand.
To further disabuse the OP of a misconceived notion, this isn't just "how many ways are there to tie your shoes". This is trying to work out a rational system of knot classification.
The key thing to realize is that knot theory applies to a lot more than untangling rope. If you use the right assumptions and definition, certain problems, which have nothing to do with rope, can be modeled as knot problems. If we could solve/simplify knot theory, we are this much closer to solving a range of related problems. N
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>>If you use the right assumptions and definition, certain problems, which have nothing to do with rope, can be modeled as knot problems
thank you posting clearly, I can never figure out what some of these theory's do, but just as soon as you posted what you did, I completely understood the basics of game theory, I'm off to learn more.
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In other words, don't be a Palin. No one likes a Palin.
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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???
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Yeah, and maybe then we wouldn't have the tools to even consider solving knot theory problems and we could go back to ending wars and eliminating human suffering. Stupid, Euler.
Also, why isn't is spelled Oiler? Was Euler some sort of anti-drilling-save-the-tress nut? Didn't he know how important oil is to our country?
Re:The hardest math (Score:5, Insightful)
Re:The hardest math (Score:5, Insightful)
Hard problems are only hard because we're using the wrong tools.
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Hard problems are only hard because we're using the wrong tools.
Hard problems are only easy if you take the script kiddie mentality of using a tool that hides the real complexity instead of understanding what's going on. Easy problems only exists in lab experiments and simplified models, reality is actually too complex most of the time. Even a basic question like "How's the weather tomorrow?" is in fact a very complex problem. Anything that involves people usually is too, "Why do people buy product X?" is a complex combination of marketing, pricing, functionality, psych
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I spent 15 years of my life in physics of proteins. Theory of knots in protein folding is nothing more than fancy mathematical excursion (though knots do matter, in very simple form). The importance of "theory" in those sciences is way overblown. It was
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And I have spent 5 years of my life on the topology of proteins. It is not quite true to refer to "knots" when talking about proteins, as Professor Taylor has shown (http://www.nature.com/nature/journal/v406/n6798/full/406916a0.html) that only a few proteins are actually 'knotted'.
However, mathematical theory of tangled strings is as important as simulations. Estimating the total number of folds, for example. More than just a fancy excursion - but maybe not to your taste?
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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.
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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.
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
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
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then you are an idiot.
programming solutions often can use mathematics theorems to produce very efficient algorithms.
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.
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From the article:
The payoff from such work may be profound. Knot Floer homology has higher-dimensional analogues that can reveal the structures of three- and four-dimensional spaces, and it is expected that Khovanov homology does as well. Four-dimensional spaces have been especially difficult to understand. Higher-dimensional spaces have enough room that complications can work themselves out, and lower-dimensional spaces are so tight that complicated behavior canâ(TM)t emerge in the first place, but in four dimensions, almost anything can happen. "Understanding four dimensions would be especially exciting," OzsvÃth says, "because thatâ(TM)s the world we live in."
In short, science speaks the language of nature which is maths. A greater understanding of four dimensional space is fundamental to the advancement of science. The work here is deep and profound in a way that is not readily apparent, but is essential in our advancment of knowledge and science.
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Surely there's only one? Sure there are probably infinite ways to create a knot that would be a bugger to get out, but that's not really tying shoelaces is it.. the whole point is they come undone when you pull them, and universally this is done only one way.
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At some point they may just go from saying "our method cannot distinguish between these two knots" to saying "our method has proven that these two knots are the same, on a fundamental level". When that happens, you're screwed.