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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!"
That guy must be pretty old
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.
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.
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.
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.
You could have worked it out with a pencil.
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.
Not to mention pilots.
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