Today, an engineering student voices a concern. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Michelle Ramotowski tells how her teacher finishes off a proof with the words, "Problem solved!" He sets the problem up, draws a diagram -- then, while the class waits for equations, he abruptly says, "Problem solved!" But he's only uttered words. Mere logic, she tells us, seems out place in this world of science and engineering. How can a solution lie in words alone!

She asks us to look at cooling fins, for example -- the kind you see on radiators, at home or in your car -- the kind that surround your overheated computer chip. Those sheet-metal plates help a hot object shed heat by increasing its surface area.

In 1926 the great German engineer Ernst Schmidt wrote about fins. He argued that a certain optimal shape would give the best cooling for a given amount of metal. He wrote no equations. He simply likened the flow of heat in a fin to the flow of water in a pipe. His physical insight led him to a simple parabolic cross-section. The argument was one paragraph long.

During the Sputnik years, we engineers grew far more formal and mathematical. By 1959 we were no longer willing to hear Schmidt's simple reasoning; and, in 1959, R.J. Duffin wrote:

It seems that the literature does not contain a proof of the Schmidt criterion. Schmidt did advance an intuitive argument, but it is not convincing.

Duffin wrote six pages of the calculus of variations. He finally convinced himself Schmidt'd been right all along. So why had Schmidt felt no need to turn mathematics on this problem? Well, in 1926 the logic of physical processes was not yet out of style.

Ramotowski thinks Schmidt offers a lesson to us all -- to her faculty and to her fellow students. "When I try to explain technical ideas to non-engineers," she says, "my equations draw blank stares. When I use pictures and analogies, the person understands. We get so caught up in calculation that we forget to step back and see the picture whole."

So she takes us back to the fin problem. In 1974, C.J. Maday wrote a new mathematical model for the optimal fin shape. He got the same overall form Schmidt did, but with a wavy surface ending in a tip whose shape was simply silly. "Sure, the proof is beautiful," she says, "but the thing obviously wouldn't work."

When you write mathematical models, you simplify problems and narrow their scope. That's a good thing to do until you give away major pieces of reality -- until mathematics replaces mental fight instead of joining it. The blind men trying to describe an elephant learned a great deal by looking at parts separately -- by writing equations for the trunk or the tail. But, in the end, we have to step back and see the elephant whole. Only when we do, can we ever really say -- "Problem solved!"

I'm John Lienhard at the University of Houston, where we're interested in the way inventive minds work.

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Schmidt, E., Die Wärmeübertragung durch Rippen. Zeitschrift des Vereines Deutscher Ingenieure, Band 70, Nr. 28, 26 Juni 1926, S. 885-889.

Duffin, R. J., A Variational Problem Relating to Cooling Fins. Journal of Mathematics and Mechanics, Vol. 8, No. 1, 1959, pp. 47-56.

Maday, C.J., The Minimum Weight One-Dimensional Straight Cooling Fin. Transactions of the ASME. Journal of Engineering for Industry, Vol. 96, No. 1, 1974, pp. 161-165.

For an introductory discussion of cooling fin design, see J. H. Lienhard IV and J. H. Lienhard V, A Heat Transfer Textbook. 3rd ed., Cambridge, MA: Phlogiston Press, 2004, Click here for a free copy., Section 4.5.

This episode was conceived and drafted by Michelle Ramotowski, a student in the Mechanical Engineering Department at UH. N. Shamsundar, UH Mechanical Engineering Department, contributed significant counsel.