# Mathematician's Dog Knows Calculus

• Star

— A mathematician in Michigan has a dog that can do calculus.

No kidding.

Tim Pennings, associate professor of mathematics at Hope College in Holland, Mich., discovered the talents of Elvis, his Welsh Corgi, while tossing a ball into Lake Michigan a couple of years ago. Elvis's performance was so compelling that Pennings published an article describing the dog's mathematical skills in the May issue of The College Mathematics Journal.

It's all pretty neat, Pennings says, because it shows that advanced math really does have practical applications in life, even if you happen to be a dog.

It so happens that Pennings' visit to the shores of Lake Michigan followed a class in which he had been trying to help his students grasp a basic problem in calculus.

Saving Tarzan

He was using a classic Tarzan-Jane illustration in which Jane gets stuck in the quicksand and Tarzan has to get to her in time to save her life. Of course, being a politically correct chap, in Pennings' problem it is frequently Tarzan who's stuck in the quicksand, needing a bit of help.

"So Tarzan is in the quicksand, and Jane is across the river and down the bank a ways, and she's got to get to him as quickly as possible," Pennings says. "She can run at a certain speed, and she can swim at a certain speed, which is obviously slower than she runs, and the question is what's her best strategy for getting to Tarzan in the quickest amount of time?"

That's actually a pretty basic problem in calculus, which is often used to find the maximum and minimum values of things, like finding the quickest route to go from Point A to Point B. The shortest route is not always the quickest, as Jane would learn if she plunged into the river and swam straight toward Tarzan. The quickest way, Pennings says, is for her to run part of the way to a point closer to Tarzan, and then jump into the river and swim across.

But where is that point?

That's the question his students were supposed to answer, and Pennings drew curved lines on the blackboard representing the variations in swimming and running, and the time required for each, and where the lines intersected, showing the best route.

He was thinking about all that heady stuff one day when he tossed a tennis ball into Lake Michigan for Elvis to fetch. As he watched his short-legged friend bound down the beach, he was astonished by Elvis's path.

"I thought, man, that's exactly what I drew on the board," Pennings says. "He's doing the very same thing."

Quick Ball Fetch

He says he could tell by the look in Elvis's eye — something only another dog lover would understand — that the pooch was determined to get that tennis ball in the shortest time possible. He didn't jump into the water and take the shortest course. Like Jane, he ran part of the way, and then picked his spot and jumped in and swam to the ball.

The next day Pennings returned to the same place with one of his students, and together they clocked Elvis's speed as he swam through the water and dashed along the beach.

"Then on another nice day I went there all by myself [along with Elvis] and thought, 'how do I figure this out?'" Pennings recalled. He ended up with a long tape measure stretched along the beach "at the interface of the water and the sand," and a screwdriver stuck in his back pocket.

"I threw the ball off into the water," he says, and Elvis took off down the beach, searching for the fastest route, with Pennings in hot pursuit.

At the point where Elvis dove into the water, Pennings stuck his screwdriver into the sand, grabbed his tape measure and followed Elvis to the ball. That gave him the measurements that he needed, and he already had Elvis's swimming and running speed.

Returning to his math lab, Pennings plotted all those points on a graph, and astonishingly, "it turns out that all the choices he made were right in line, or very close, to the optimal choice," he says.

Innate Math Genius

Subsequent expeditions to the shores of Lake Michigan verified the findings, and Pennings was able to build a case strong enough to interest a professional math journal.

Of course, the professor doesn't think Elvis was really doing calculus, at least not in the traditional way. But somehow, innately, he was achieving the same result.

Pennings figures a lot of dogs probably do the same thing, but oddly enough, he suggests, their masters may not be thinking about calculus when they throw a stick or a ball into the water for their dog to fetch.

Elvis is now on sort of a canine lecture tour, helping Pennings explain calculus to students of all ages.

Advanced math does have practical applications, Pennings tells the students. If you end up as an industrialist, he tells them, and you manufacture a certain item, you will need to come up with a formula that will tell you how many you need to manufacture to maximize your profit, or minimize your cost.

And as Elvis would say, if he could speak English, that's calculus.

That's all pretty impressive, but could he really solve a basic problem that baffles many a calculus student? Could he, for example, determine the hydrostatic pressure at the base of a dam?

Not a chance.

But if he ever does, we better be ready to play that old theme song from The Twilight Zone.

Lee Dye’s column appears weekly on ABCNEWS.com. A former science writer for the Los Angeles Times, he now lives in Juneau, Alaska.