Forgot your password?
typodupeerror
Math Education

Mathematician Solves a Big One After 140 Years 144

Posted by kdawson
from the works-with-holes dept.
TaeKwonDood notes that ScientificBlogging.com has just written about a development in applied math that was published last year. "The Schwarz-Christoffel transformation is an elegant application of conformal mapping to make complex problems faster to solve. But it didn't do well with irregular geometries or holes, so it simplified too much for a lot of modern-day mechanical engineering applications. 140 years after Schwarz and Christoffel's work, a professor at Imperial College London has generalized the equation. MatLab users rejoice!"
This discussion has been archived. No new comments can be posted.

Mathematician Solves a Big One After 140 Years

Comments Filter:
  • wow (Score:5, Funny)

    by Anonymous Coward on Monday March 03, 2008 @10:04PM (#22631348)
    That guy must be pretty old
  • Math Forfront (Score:5, Insightful)

    by Bananatree3 (872975) on Monday March 03, 2008 @10:08PM (#22631374)
    It always amazes me how applicable math becomes hundreds of years after it's written. Think if Maxwell's equations, Newton's equations, Einstein's equations. Fluid Dynamics equations were probably pioneered well before they were applied to human machines. Modern-day aircraft would not operate without their understanding. Where the math goes, human technology will probably soon follow.
    • Re: (Score:3, Insightful)

      by siride (974284)
      I think they would operate. The universe doesn't need to know the math. It Just Works (TM).
      • Design (Score:3, Funny)

        by Bananatree3 (872975)
        of course pilots don't need to know the math behind why their plane works. I sure hope the designers of the planes knew their math! Without them the planes wouldn't work.
        • Re:Design (Score:5, Insightful)

          by ceoyoyo (59147) on Monday March 03, 2008 @11:27PM (#22631890)
          Designers designed planes long before they could work out the math. They experimented a lot. The math lets you make things faster, cheaper and gives you ideas for new designs. I wouldn't fly in anything based solely on the math though.
          • Re:Design (Score:5, Insightful)

            by Bananatree3 (872975) on Monday March 03, 2008 @11:41PM (#22631986)
            I agree. The Wright Brothers knew only some basic math and mostly built their airplane through ingenious yet fairly simple experimentation.

            That's why I emphasized modern-day aircraft. Designing a 777 or the new 7E7 off pure experimentation would take insanely more amounts of time and money. Math makes it a LOT easier, and its probable all turbine-driven commercial craft wouldn't exist at their current efficiencies without math being in the design process. Laugh all you want about their gas-guzzling reputations, but it would be interesting to see someone design such a sophisticated aircraft without advanced math.

            • Re:Design (Score:5, Interesting)

              by ceoyoyo (59147) on Tuesday March 04, 2008 @12:31AM (#22632284)
              It makes it cheaper, but you can certainly have sophisticated turbine aircraft without the math. We've only had the computers to make a respectable stab at simulating airflow over a reasonably complex wing recently. It's great as a design aid, and invaluable as a tool for understanding, in the abstract, but the real world is often too complex for our computational capabilities. Surprises crop up all the time. The A380 wing for example. Its probably the modernest and advancedest turbine-driven commercial aircraft wing (at the moment). The wing in practice isn't as efficient as it was supposed to be. It also failed its strength certification the first time around.

              In most engineering applications the math is a nice tool to let designers do a bunch of experimenting inside the computer before they have to move on to real world testing. We're not at the point yet where math is more important than experience and experiment. Not just aircraft design. I work in medical imaging and there are no shortage of ideas where the (idealized) math works great, the simulations are wonderful, but the idea doesn't survive first contact with patient data.

              • first contact with the medical engineering?

                You couldn't have tomography without computer assistance, true, but you have lots of people going around with radiation burns from improperly calibrated X-ray equipment.
                • by ceoyoyo (59147)
                  Thus the extensive testing. We still can't accurately calculate the absorbed radiation dose for a patient, only approximations, so guidelines are based on some simulations but also a lot of experiment. Plus a hefty safety factor. Accidents are actually remarkably rare, because of the testing required, but when they do happen they can be pretty horrific.
              • Re: (Score:3, Informative)

                I am a designer for a large gas turbine engine manufacturer, and I have to agree that there is still a lot that we just don't understand well enough or can't model adequately. Combustion noise, liquid atomization, fatigue/creep interaction, etc. We do all kinds of FEA and CFD analysis, but still spend tens of millions of dollars on testing to back up those simulations.
              • Re: (Score:2, Insightful)

                by Bloodoflethe (1058166)

                It makes it cheaper, but you can certainly have sophisticated turbine aircraft without the math.
                Your opening argument has no supporting statement whatsoever. You don't refer to a single instance of making sophisticated aircraft without math in this whole post. When you make a radical statement such as that, you really should back yourself up with a source. But, then again, this *is* slashdot
                • Re:Design (Score:4, Informative)

                  by ceoyoyo (59147) on Tuesday March 04, 2008 @11:23AM (#22636658)
                  Well, the 757 was designed in 1983. Certain versions of it have a reputation for being very fuel efficient. The U2 and SR-71 were designed and built in the 40s and 50s, and the SR-71 is still the fastest aircraft to take off under its own power. The H-4 Hercules was designed and built in the 40s and has the largest wingspan and height of any aircraft in history. The 747, one of the most successful commercial aircraft, was designed during the 60s.

                  So it depends what you mean by "math." The Wright brothers undoubtedly needed to add and subtract measurements to build their plane. That's math. Those designers in the 50s and 60s used pencils, slide rules and tables to work out some equations to help guide them (there was some talk of using the new electronic computers, but aircraft designers weren't overly enamored of them). The big aircraft manufacturers started developing 2D computational fluid dynamics software in the 70s, and two major packages were developed in the 80s.

                  So what about today? Well, you won't find a test pilot who's willing to fly a new design that hasn't been tested in a wind tunnel. There's no way I would fly on an aircraft that hadn't been tested in real flight, unless I was being paid (and trained) as a test pilot. Aircraft companies spend billions on wind tunnels. It seems even today the math is awfully useful but it's no substitute for putting an aircraft in an airstream and seeing what happens.

                  Sources:
                  http://en.wikipedia.org/wiki/Computational_fluid_dynamics [wikipedia.org]
                  Cosner, RR and Roetman, EL, "Application of Computational Fluid Dynamics to Air Vehicle Design and Analysis", IEEE Aerospace Proceedings, 2: 129-42 (2000).
            • Re:Design (Score:5, Funny)

              by h4rm0ny (722443) on Tuesday March 04, 2008 @02:33AM (#22632950) Journal

              Designing a 777 or the new 7E7 off pure experimentation would take insanely more amounts of time and money.

              Not to mention pilots.
              • by MmmmAqua (613624)
                I wish I had mod points. That made me spray pad thai all over my (employers) LCDs. Good on ya. :)
          • by Ngarrang (1023425)

            Designers designed planes long before they could work out the math. They experimented a lot. The math lets you make things faster, cheaper and gives you ideas for new designs. I wouldn't fly in anything based solely on the math though.
            Fancy math is what keeps the F-117A from falling out of the sky.
            • by ceoyoyo (59147)
              There was a LOT of experimentation involved in designing the F-117A. The wind tunnel and flight test data from earlier test designs was used extensively in developing the F-117A, including building a simulator so the test pilots would know what to expect. They didn't have a copy of X-Plane back then.
        • Re: (Score:3, Insightful)

          by Joe Snipe (224958)
          Tell that to the dragonfly
      • Sure the forces, etc, that enable such things to work would be there. But, that means nothing when it comes to us building something that takes advantage of such forces, etc. For that to happen, it takes math and science i.e. understanding.
      • by kalirion (728907)
        Kind of like an operating system doesn't need to know the machine language?
    • Re:Math Forfront (Score:5, Interesting)

      by HungSoLow (809760) on Monday March 03, 2008 @10:12PM (#22631394)
      There is a saying that goes something like "for every new discovery in math, a new field of science begins".
      • Re: (Score:3, Insightful)

        by ceoyoyo (59147)
        A saying in math.

        Reality is more like, for every discovery in science, a mathematician developed the relevant math in the abstract a hundred years earlier.

        Not as catchy, I know.
        • by sconeu (64226)
          Semi related: a sig seen on /.:

          "An interesting anagram of "BANACH TARSKI" is "BANACH TARSKI BANACH TARSKI".

          Apparently, the B-T theorems can be used to describe quark behavior.

          • by ceoyoyo (59147)
            There are hints that the non-trivial zeros of the zeta function, besides being related to the distribution of the prime numbers, also form an operator that describes a particular quantum mechanical system. You wouldn't say that the zeta function created quantum mechanics, but it might come in handy at some point in the development of QM.
    • Re: (Score:3, Interesting)

      by nwf (25607)
      I think that rather than math becoming applicable, it actually enables discovery and enables people to think about problems. Without many seemingly uselessly arcane topics, we'd be back in the 1900s. Calculus comes to mind. Heck, physics these days seems to be nothing more than experimental mathematics with string theory and the like.
      • Re:Math Forfront (Score:5, Insightful)

        by 644bd346996 (1012333) on Monday March 03, 2008 @10:31PM (#22631544)
        Calculus is one of those things that was created more or less with a real-world application in mind (ie. physics). A better example would be how abstract algebra (in specific, group theory) has recently found application in quantum mechanics. Both fields have been around for quite a while, but they only recently connected.
        • by colfer (619105)
          Coding theory, crypto, general relativity... there are tons of examples where the math(s) anticipated the physics by decades or more. But solid applications keep math healthy too.

          We used to have this saying in the pure math dept.: hey does this have any applications? Yes, it has applications to number theory!
        • by nwf (25607)
          Calculus also birthed differential equations, which are used all through engineering, and even the Fourier transform, without which we wouldn't have cell phones or MP3s. But, abstract algebra is a good example, but I haven't used it much since college. And number theory is the basis of modern cryptology.
        • Calculus is one of those things that was created more or less with a real-world application in mind (ie. physics).

          That was certainly Newton's intention. Leibniz had other goals in mind.

        • by hawkfish (8978)
          Considering the Greek attitude towards practical applications of mathematics, I sort of doubt Archimedes invented integral calculus with physics in mind.
      • by colfer (619105)
        ...and really really big frocking machines.
      • Re:Math Forfront (Score:5, Insightful)

        by zippthorne (748122) on Monday March 03, 2008 @10:35PM (#22631562) Journal
        Except, Calculus, specifically, was invented by the same guy who used it to basically describe classical physics. And he also proved all of his theorems using geometry, since the new-fangled calculus might not be acceptable for proofs just yet, having only just been invented, by him.

        The point is, how can you separate the invention of calculus from his work in classical physics? They were obviously developed hand-in-hand.
        • Re:Math Forfront (Score:4, Informative)

          by bjorniac (836863) on Monday March 03, 2008 @10:49PM (#22631652)
          Really? Leibniz invented physics?

          OK, I know what you're saying, but really, Newton takes too much credit here. In his early work he even credited Leibniz then in a later edition of his work removed the statement.
          • Re: (Score:3, Insightful)

            It's a good thing this argument is still going on since they both discovered/invented calculus pretty much independently, perhaps with some borrowing between the two. Newton started before Leibniz, Leibniz did a better job making it useful, and Newton definitely did more with it. The both invented it, end of story. Seriously, this is over two centuries old, let it die.
            • by KevinKnSC (744603)
              I agree. Now, with that out of the way, let's get back to the Cardano-Tartaglia debate. That's where the real action is.
            • It's a good thing this argument is still going on since they both discovered/invented calculus pretty much independently
              If only Newton had patented his ideas, then there would be no question!!!
          • by pbhj (607776)
            Can you upload a photo of that page ... I'd love to see it! Sounds like folklore to me.

            I've always considered them to have been largely independent not least because of the different notations adopted. It was calculus's time, if not Newton or Leibniz some other genius ...
      • by amplt1337 (707922)
        Heck, physics these days seems to be nothing more than experimental mathematics with string theory and the like. ...and that's precisely the kind of physics that may as well be philosophy, for all the science it actually does.
    • "It always amazes me how applicable math becomes hundreds of years after it's written."

      All mathematics is descriptions of geometry, hence why math is applicable. You have a sphere: How are you going to describe it? Math is just an abstract representational system to describe structure, shapes and relationships.
      • Once you get past 3-dimensions, you can have mathematical concepts that you can't associate with any shape our minds are capable of accurately imagining.

        You can also use mathematics to manipulate infinitely large numbers, or irrational numbers - what shape do they represent?

        Or consider a very simple form of substitution algebra:
        a = pq
        x = by
        qb = ag
        You can prove ax = ppqgy. How would you represent that geometrically?

        Your definition is too limiting.
        • Once you get past 3-dimensions, you can have mathematical concepts that you can't associate with any shape our minds are capable of accurately imagining

          I hear this a lot and am not sure that I agree... I can, quite clearly, picture a hypercube in my mind. I can't describe it verbally (or at least, not without starting well but finishing lamely with a "sort of, the other direction to those three"), draw it on paper or model it in clay, but I can definitely picture it clearly.
          The first time, as a young child, that I was introduced to the idea, I really couldn't picture it at all, but then I just became more and more accustomed to the idea and could event

          • Can you picture any number of dimensions for objects, or do you have a limit? I would think that eventually, you would hit a point where you can make mathematical models about the item but your brain isn't capable of generating an image of the shape.
        • I should have mentioned this in my last post (just above)... I forgot to mention that you are still actually completely correct though that we can't accurately model it with physical geometry, nor are we able to explain it sensibly and accurately without the language of mathematics, regardless of whether it's possible to mentally imagine it. So your point is still 100% valid.
    • Re:Math Forfront (Score:5, Insightful)

      by pclminion (145572) on Monday March 03, 2008 @10:49PM (#22631662)

      It always amazes me how applicable math becomes hundreds of years after it's written. Think if Maxwell's equations, Newton's equations, Einstein's equations. Fluid Dynamics equations were probably pioneered well before they were applied to human machines. Modern-day aircraft would not operate without their understanding. Where the math goes, human technology will probably soon follow.

      It's often debated whether mathematics is invented or discovered. I think the question is irrelevant. Mathematics is clearly a human endeavor. Whether it has some deeper meaning outside of human existence is not something we can even address, seeing as we can never step outside our human condition. But it is indisputable that mathematics has allowed us to move far beyond the boundaries of any other physical organism that we yet know of. Whether it's "real" or not, it is certainly real in the context of our own existence. The philosophical arguments between mathematicians and physicists are petty at best. Ultimately, all new math seems to find application in the physical world. We should not be surprised, given that we are physical beings.

      I feel pride, not in humanity, but in the universe itself, that it has the capacity to create physical beings which are capable of comprehension, at least at a basic level, of the true nature of reality. It may be colored by our nature, but the triumphs of modern science, in particular nuclear energy, show that we may actually be aware of some fundamental truth. The law of mass-energy equivalence can be demonstrated through purely geometric arguments -- you need not even understand calculus in order to grasp the math. We have grasped the power of stars. That proves something about us, but I am not sure what.

      • by greg_barton (5551) *

        ...seeing as we can never step outside our human condition.

        Genetic engineering and/or cybernetics, enabled by mathematics, may well change that.
        • by melikamp (631205)

          And then, may be, one day, math will finally calculate the exact limit to the Human Pride. Or may be the whole sum of it will just diverge to +00.

      • Re:Math Forfront (Score:5, Insightful)

        by ceoyoyo (59147) on Monday March 03, 2008 @11:58PM (#22632080)
        We should also not be surprised since we construct math from its basic axioms to make sense to us logically - i.e., to work the same way reality does.

        The really amazing thing is that the universe appears to respect our ideas of logic.
        • by 12357bd (686909)

          The really amazing thing is that the universe appears to respect our ideas of logic.
          Well, only in the part of the universe that we already know, not that much in this light.
        • Re: (Score:2, Interesting)

          by 3D-nut (687652)
          If you haven't already, you might want to read Eugene Wigner's essay, on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Here's one link: http://nedwww.ipac.caltech.edu/level5/March02/Wigner/Wigner.html [caltech.edu]
          • by ceoyoyo (59147)
            Hey, thanks. I read it somewhere before, but there's nothing like having a copy. A link to the original journal too.
        • It's not that amazing. A brain that wasn't reality-based wouldn't evolve in the first place.
          • by ceoyoyo (59147)
            That's one possible explanation. Except our brains only have to deal with a very small domain within the universe. In fact, our brains don't work very well when we aren't talking about light, large things that move slowly. Our intuition doesn't cover relativity or quantum mechanics very well. But the math does. Our ideas of logic, developed to deal with problem solving in our limited environment, still works in environments that we didn't evolve to understand.
            • Logic isn't generalized problem solving though. It's a lot more specific. Logic is a method of talking about equivalencies of statements. A few axioms about lines and points are equivalent to all of Euclid's proofs, etc.

              So it's a very understandable mental module -- being able to say whether two natural language statements are equivalent or not -- that gives us all of math and physics.
              • by ceoyoyo (59147)
                It's the steps in between: A -> B. Why does the -> work? A and B aren't equivalent, the truth of A implies that B must be true as well, and there are rules for what's kosher getting from A to B. Why? And why can those rules, combined with a few reasonable starting assumptions, take you so far?

                You're right, our brains must be wired to recognize reasonable rules for -> in normal circumstances, but why do those rules continue to hold outside of our everyday experience? The picking and choosing sc
        • The universe does not at all respect our idea of logic. Our ideas of logic change to fit with observations.

          We only understand gravity because we observe it by falling on our diapered butts as babies. Therefore gravity becomes part of our logic.

          Heavier than air flight was impossible (our logic told us), until proven otherwise and we had to modify our logic.

          Going faster than 60mph, then 100mph, then sound would certainly kill people, until it was done.

          Our logic tells us the world is flat, etc etc.

          Stuff like q

          • by ceoyoyo (59147)
            You're confusing logic with intuition. Our intuition tells us that heavier objects should fall faster than lighter ones. Remove the air and we see that it isn't always true. Our intuition has misled us.

            Logic, on the other hand, always seems to work. If your theory doesn't work you examine it for errors and check your assumptions. You don't go back and wonder whether logical deduction has failed.
      • by mgblst (80109)
        Mathematics is clearly a human endeavor.

        Are you suggesting that, in the case that there is other life out there, that they won't come up with the same mathematical system that we have? Of course not.
        • by pclminion (145572)

          Are you suggesting that, in the case that there is other life out there, that they won't come up with the same mathematical system that we have? Of course not.

          That conclusion is unjustified. A physical being which is incapable of distinguishing "numbers" is obviously not going to have any sort of mathematics, or logic for that matter, even remotely close to ours. If you think math is obviously universal, you clearly haven't taken hallucinogens before.

      • Go and look at the areas of Physics that don't have a sound basis in Maths, there are the ones that don't work yet, don't actually predict anything, and are often put down as speculative.... the various Sting theories spring to mind...

        Whereas many new theories in Physics that were based on well known maths (or were found to be...) very quickly became applicable in the real world and are now used in everyday life not just in physics labs or physicts heads ...experiemt is all very well but unless you know w
      • Math has two distinct aspects.

        First there is math as is relates to physics principles. 1 + 1 must equal 2. In a classical wphysics world there is no getting around that. Arithmetic, Pi, e and a few others are discoverable math principles.

        But, second is how we as human beings understand math, this is invented. There is no fundamental reason why calculus is as it was developed. Caculus represents our understanding of math and is an invention of convinience.

        Remember, all math COULD be done with basic arit
    • by Dutch Gun (899105)
      Quaternions, first described in the mid 1800s, were essentially a solution without a problem until they became relevant for computer-generated animation and graphics. Until then, I believe they were mostly just considered a mathematical curiosity.

    • by jd (1658)
      Based on these notes [princeton.edu], placed on a public web server by one of Princeton's greatest mathematical minds, where would humans go?
  • Article text (Score:4, Informative)

    by melikamp (631205) on Monday March 03, 2008 @10:49PM (#22631654) Homepage Journal

    The article [ic.ac.uk] is available at the author's website [ic.ac.uk].

    As far as I can tell, the original result provided a conformal map [wikipedia.org] from a disk onto a polygon. Prof. Crowdy extended this result to provide a map from a disk with circular holes poked in it onto a domain with polygonal holes. Why is it useful? I am sure someone in the applied camp would know.

  • by l2718 (514756) on Monday March 03, 2008 @11:02PM (#22631730)

    Read the paper. This is not the first S-C formula for multiply connected regions. The claimed "key result" is a formula for a case where a formula is already known. More work will be needed to a adapt the MATLAB technology from singly- and doubly-connected regions to multiply connected regions.

    This paper seems to be part of ongoing work by a small community and is probably useful, but it's not a major mathematical breakthrough -- more of an incremental step. Small technical improvements in one field of mathematics shouldn't make up a slashdot story. Just because someone put "140 year old" in the press release doesn't mean it's really important. A math story belongs on /. when a big result is announced -- on the level of Poincare's Conjecture, or the Modularity Theorem.

    • by melikamp (631205)

      Does it really feel like there is too much math on Slashdot? Only reporting the likes of Poincare's Conjecture would be similar to only reporting "P=NP" and "computer passes full Turing test" for computer science.

      • by l2718 (514756) on Tuesday March 04, 2008 @04:15AM (#22633428)

        Does it really feel like there is too much math on Slashdot?

        No, it feels like there is the wrong math on Slashdot. What is needed are stories explaning accessible mathematics to a general audience. Not needed are stories about technical advances in mathematics. Two years ago there was a big hoopla about the calculation of the unitary dual of the split real form of $E_8$, which was a more important result and still completely irrelevant to the general public and impossible to explain even in the vaguest terms. There exists blogs by mathematicians where new math results are discussed. Slashdot should find stories which explain ideas of math, and report the occasional genuine breakthrough.

        For CS, which is closer to the readership than Math, the bar should be lower. Deterministic poly-time primality testing was reported; a faster matrix multiplication algorithm, or even a faster factorization algorithm should be reported even if the details of the algorithm will not be reportable.

        • by KefabiMe (730997)
          Might I remind you, that Slashdot is "News for Nerds. Stuff that Matters." As a mathematician, I definitely fall into the "Nerds" category. While this might not be "Stuff that Matters," a proof of something like the Reimann Hypothesis is way above the heads of an everyday person. However, I would be outraged if Slashdot somehow failed to run the story on the front page.
      • Re: (Score:2, Interesting)

        I like the math articles on here. Usually I'm reduced to a "eh??" (I've ~30 credits of college math but most of the interesting stuff is well beyond that) but when someone here takes a significant discovery and breaks it down so I can understand it ... that's one of the things I most love about /.
    • by siwelwerd (869956)
      Not to mention the linked article is so poorly written and lacking in details that after reading it I had no idea what had actually been shown. As to why this is important, I'm no analyst so I'm not going to read the entire paper, but it appears that he's made improvements to computing such a conformal map, which was previously more computationally difficult.
    • by ceoyoyo (59147)
      When I saw the headline I remembered when Fermat's last theorem was solved and immediately thought of the Poincaire conjecture and the Reimann Hypothesis. I was disappointed.
    • Re: (Score:3, Informative)

      by gardyloo (512791)
      Indeed. See this 1956 paper: http://links.jstor.org/sici?sici=0002-9947(195605)82%3A1%3C128%3AOTCMOM%3E2.0.CO%3B2-P [jstor.org] (warning: links to only an abstract on JSTOR).

            Conformal mapping is pretty easy to explain to a lay audience (no, not necessarily hookers); the original article did a horrible job.
  • by syousef (465911) on Monday March 03, 2008 @11:27PM (#22631892) Journal
    I knew I could have scored better if there were no time limit!

    Miss, I'd like 140 years to finish my paper!
  • by Curl E (226133) on Tuesday March 04, 2008 @06:18AM (#22633894)
    Should the rejoicing be limited to users of proprietry linear algebra systems?

You had mail, but the super-user read it, and deleted it!

Working...