Ivars Peterson's MathTrek
In 2003, mathematician Tim Pennings of Hope College in Holland, Mich., revealed to the world that his Welsh corgi, Elvis, appears to be solving a calculus problem when finding the optimal path to fetch a ball. In this case, optimal path means minimizing travel time.
Depending on the dog's running and swimming speeds, the strategy that Elvis follows appears to minimize the time that it takes to get to the ball. Indeed, Pennings found by experiment that Elvis performs in a way that closely matches a calculus-based mathematical model of the situation.
"It seems clear that in most cases Elvis chose a path that agreed remarkably closely with the optimal path," Pennings argued in the May 2003 College Mathematics Journal.
Now, several other researchers have weighed in on the question of what sort of calculations dogs may do to reach their goals.
In the January College Mathematics Journal, Pierre Perruchet of the University of Bourgogne and Jorge Gallego of Robert-Debre Pediatric Hospital in Paris contend that the model chosen by Pennings assumes that the dog knows the entire route in advance in order to minimize the total duration of travel. Instead, they say, a dog optimizes its behavior on a moment-to-moment basis.
Perruchet and Gallego worked with a female Labrador named Salsa, who, like Elvis, apparently chooses the optimal path when playing fetch along a lakeside beachin this case, near Nimes, France.
The researchers suggest that a dog playing fetch chooses at each point in time the path that allows it to maximize its speed of approach to the ball.
However, for this alternative model to work, a dog must be able to estimate accurately its speed of approach at each moment and to have a general awareness of its swimming speed before entering the water. Perruchet and Gallego argue that dogs and other animals do have such motion detection capabilities.
On the other hand, Pennings insists that Elvis appears to make global decisions rather than instantaneous decisions when retrieving a ball.
The following experiment suggests why. "Playing fetch with Elvis, I decided to throw the stick while standing in the water, about 10-12 feet from shore, and with Elvis right beside me," Pennings reports. "When I threw the stick in a path parallel to the beach, Elvis swam in to shore, ran along the beach for a sizeable distance, and then dove back into the water to retrieve the stick."
"Thus," he adds, "in swimming to shore he was not acting to minimize his distance to the stick as quickly as possible. Instead he did in fact apparently make a 'global' decision form the outset as to what path would get him to the stick most quickly."
In the same issue of the College Mathematics Journal, mathematician Leonid Dickey of the University of Oklahoma proposes an extension—a strategy that dogs might use if they were initially not at the water's edge but standing some distance from the shore. This becomes a problem in the calculus of variations.
Dickey then asks how a dog would respond if the soil properties (such as density and water content), and hence the running speed, changed gradually. But he presents no experiment data. Perhaps he doesn't own a dog.
In the meantime, Elvis (full name Elvis Bogart Wales) has gone on to bigger and better things. A year ago, he was awarded an honorary degree "Litterarum Doctoris Caninarum" from Hope College. He even made a guest appearance in Keith Devlin's new book, The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs).
Devlin, K. 2005. The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs. Thunder's Mouth Press. See http://www.mathinstinct.com/excerpt.html.
Dickey, L.A. 2006. Do dogs know calculus of variations? College Mathematics Journal 37(January):20-23.
Gallego, J., and P. Perruchet. 2006. Do dogs know related rates rather than optimization? College Mathematics Journal 37(January):16-18.
Pennings, T.J. 2003. Do dogs know calculus? College Mathematics Journal 34(May):178-182. Available at http://www.maa.org/features/elvisdog.pdf.
Peterson, I. 2004. Dog does calculus. Muse 8(January):27. Available at http://www.sciencenewsforkids.org/pages/puzzlezone/muse/muse0104.asp.
______. 2003. A dog, a ball, and calculus. MAA Online (June 9).
Seely, R. 2005. Canine cuts a wide swath in math circles. Wisconsin State Journal (Feb. 15). Article.
Sohn, E. 2003. It's a math world for animals. Science News for Kids (Oct. 8). Available at http://www.sciencenewsforkids.org/articles/20031008/Feature1.asp.