Mathematician Solves a Big One After 140 Years

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!"

Comments

wow

[Anonymous Coward]

That guy must be pretty old

Re:wow

[nwf]

But can you prove it? There's got to be a limit somewhere here...

Math Forfront

[Bananatree3]

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:Math Forfront

[siride]

I think they would operate. The universe doesn't need to know the math. It Just Works (TM).

Re:Math Forfront

[HungSoLow]

There is a saying that goes something like "for every new discovery in math, a new field of science begins".

Re:Math Forfront

[nwf]

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

[644bd346996]

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.

Re:Math Forfront

[zippthorne]

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

[bjorniac]

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:Math Forfront

[pclminion]

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.

Re:Math Forfront

[moderatorrater]

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.

Re:Math Forfront

[ceoyoyo]

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.

Re:Math Forfront

[ceoyoyo]

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.

Re:Math Forfront

[3D-nut]

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

Re:Math Forfront

[SQLGuru]

What does Gordon Sumner's http://www.imdb.com/name/nm0001776/ theories have to do with anything?

Layne

Re:Math Forfront

[geminidomino]

(He did develop some interesting techniques for calculating the area under a curve, that were similar in some regards to integral calculus, but only in the sense that a longbow is similar to a modern rifle. Incredible for its time, beautifully elegant, but I know which I'd rather be using in a life-or-death situation.) What the fuck are you doing that you'd be in a life-or-death situation that depends on integral calculus?

Dude, you need new friends...

Design

[Bananatree3]

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

[ceoyoyo]

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

[Bananatree3]

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

[ceoyoyo]

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.

Re:Design

[h4rm0ny]

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

Not to mention pilots.

Re:Design

[Zed is not Zee]

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:Design

[Bloodoflethe]

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

[ceoyoyo]

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
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

[Joe Snipe]

Tell that to the dragonfly

I solved a big one this morning too

[BadAnalogyGuy]

I give credit to all the bran I've been eating lately.

Re:I solved a big one this morning too

[sakusha]

You could have worked it out with a pencil.

Re:I solved a big one this morning too

[WgT2]

Is there no limit to potty-humor?

Why must it be integrated into our lives so often?

Article text

[melikamp]

The article is available at the author's website.

As far as I can tell, the original result provided a conformal map 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.

Re:Article text

[tqft]

Only for purchase linked except this one which has more detail (under News)
http://sinews.siam.org/old-issues/2008/januaryfebruary-2008/breakthrough-in-conformal-mapping

Re:Article text

[jo42]

a) Make the square hole bigger, or, b) Put the round peg in a lathe and turn it down so that it fits in said hole.

Not quite a breakthrough

[l2718]

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.

Re:Not quite a breakthrough

[l2718]

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.

Re:Not quite a breakthrough

[gardyloo]

Indeed. See this 1956 paper: http://links.jstor.org/sici?sici=0002-9947(195605)82%3A1%3C128%3AOTCMOM%3E2.0.CO%3B2-P (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.

Re:Not quite a breakthrough

[entropiccanuck]

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 /.

High school math tests

[syousef]

I knew I could have scored better if there were no time limit!

Miss, I'd like 140 years to finish my paper!

Octave, Scilab and SAGE users rejoice

[Curl E]

Should the rejoicing be limited to users of proprietry linear algebra systems?

• Octave
  
• Scilab
  
• SAGE