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Mining Marden's Theorem

Photo: Heather Kinsman

Dan Kalman, a math professor in AU’s College of Arts and Sciences, has long been fascinated by Marden’s Theorem, which he describes as “the most amazing combination of ideas.”

In fact his paper “An Elementary Proof of Marden’s Theorem” won the Mathematical Association of America’s (MAA) Lester R. Ford Award.

Imagine Kalman’s surprise after the paper’s publication when he was contacted out of the blue by an engineer in West Virginia who wanted to use the theorem to improve mine safety.

More on that in a moment.

Kalman’s work on Marden’s Theorem is his sixth paper to win a writing award from the MAA.

The theorem also gets a mention in Kalman’s book Uncommon Mathematical Excursions: Polynomia and Related Realms (whose cover pictures the island of Maxministan lying in the Tangence Sea due west of Percent Isle).

So what is Marden’s Theorem? First of all, it wasn’t formulated by Morris Marden at all; Kalman notes that Marden attributed it to Jörg Siebeck.

An abstract of Kalman’s paper describes it this way:

“Marden's Theorem concerns the relative positions of the roots of a cubic polynomial and those of its derivative. Specifically, if the cubic has distinct non-collinear roots in the complex plane, and thus are the vertices of a triangle T, then the roots of the derivative are the foci of the unique ellipse inscribed in T and tangent to the sides at their midpoints.”

But let’s say you’re math-impaired, that you haven’t had a math class since your junior year. In high school. How would Kalman describe the elegance of Marden’s Theorem to such a hypothetically math-averse person?

“Suppose that you lost your keys and you’re walking around, looking around trying to find your keys and you can’t find them,” Kalman said. “But what you do find is somebody’s purse. And you open up this purse and inside you find a wallet and a compact and a brush and a set of car keys. And you look at the car keys and you say, ‘Gee. This person has a Toyota, too.’ And you took that car key and you put it in your door lock and it opened the door. Now wouldn’t you think that was a pretty singular experience?

“Well, that’s what this is like . . . You’ve got three points, three numbers that define something and that something tells you about two more numbers and you want to find those two other numbers. And the fact that they show up as the foci of an ellipse inscribed in the triangle is like picking up a key at random and having it fit the lock in your car.”

That ellipse inside a triangle is what interested Monte Hieb, an engineer with the West Virginia Office of Miners' Health Safety and Training, in Kalman’s work on the theorem.

“Dan was of assistance in providing me with two useful equations that helped me to automate a new procedure for evaluating rock stress measurements made with a mechanical strain gage,” Hieb wrote in an e-mail. “When it turned out that one of the equations required additional iterations to work over the full range of required angles, he was kind enough to offer to have one of his students to look into this as a school project to see if a more straight-forward calculation was available.”

Hieb explained that “ellipse constructions are used in geology and mining to express and evaluate ground stresses. This has relevant application both in mine safety and in mining economics . . . It is hoped that the simplified approach I am working on will provide another tool to mining professionals to quickly evaluate ground stress. I am using this method currently to assist in evaluating the ground stresses in a mine that has a history of periodically experiencing gas incursions through fractures in the mine floor. My thanks to Dan Kalman for his mathematical advice and assistance which aided me in automating a part of this process.”