Hope Mathematician Finds His Dog Is a Calculus Natural
Posted May 13, 2003
HOLLAND -- You may not be able to teach an old dog
new tricks, but you can get him to solve a calculus problem.
That's what Dr. Tim Pennings, associate professor
of mathematics at Hope College, realized when he took his
Welsh Corgi, Elvis, for an outing to Lake Michigan to play
fetch with his favorite tennis ball.
"Most calculus students are familiar with the
problem of finding the optimal path from point A to point
B," he said. "Standing on the water's edge at Point A, I
throw the ball into the water at Point B. Elvis runs along
the shore a portion of the way, then plunges into the lake
at a point of his choosing and swims diagonally to the ball.
By the look in Elvis's eyes, it seems clear that his
objective is to retrieve it as quickly as possible."
Pennings assumed, then, that Elvis unconsciously
attempts to find a path that minimizes retrieval time--a
problem typically solved using techniques from calculus.
But he wanted to test his theory.
So, he clocked Elvis's running and swimming speeds
and spent three hours collecting data measuring the distance
between the ball and the shore and where Elvis jumped into
the water to retrieve it. Plotting the results revealed
that Elvis almost always chose a path which is in close
agreement with the optimal path calculated mathematically.
In fact, given complicating factors--waves, the
movement of the ball in the water and others--Pennings
suggests that dogs may choose a path that is actually better
than the calculated ideal.
"Although he made good choices, Elvis does not
know calculus," Pennings said. "Though he does not know the
calculations, Elvis's behavior is an example of the uncanny
way in which nature often finds optimal solutions."
The topic was a natural for Pennings, who
specializes in mathematical modeling--the way in which
mathematics describes natural phenomena. As a teacher, he
appreciates having a new opportunity to relate his
discipline to everyday experience.
An article he has written about his work with
Elvis has been published as the cover story of the May,
2003, issue of "The College Mathematics Journal," complete
with a photo of the dog relaxing on the beach with tape
measure and ball nearby.
In addition, Pennings, accompanied by Elvis, has
also given talks on the subject at Hope and a variety of