For a Welsh corgi named Elvis, merely fetching a ball isn't enough. The dog also works out the optimal path he should take to retrieve the ball in the shortest possible time—all the better to squeeze in some extra rounds.
Welsh corgi Elvis lives to fetch. Photo courtesy of Tim Pennings.
Having established that Elvis apparently solves a calculus problem each time he fetches a ball thrown from a sandy beach into the water, mathematician Tim Pennings of Hope College in Holland, Mich., has since studied his dog's problem-solving ability when Elvis has to choose between two qualitatively different options on which path to take.
While playing fetch at the beach, Elvis sometimes starts in the water. When a ball is thrown parallel to the shore, Elvis has the option of swimming directly to the ball or heading for the shore, running on the sand, then swimming back out to the ball. If the distance to the ball is short, the best strategy is to swim directly to the ball. For longer distances, the swim-run-swim option is Elvis's best bet to minimize retrieval time, assuming that Elvis runs faster than he swims.
"Such a situation induces a bifurcation in his optimal strategy," Pennings and colleague Roland Minton of Roanoke College note in the November College Mathematics Journal. In other words, at some distance, the optimal strategy must change from swim to swim-run-swim.
The mathematicians ask, "What is the bifurcation point at which the optimal strategy changes?" And does Elvis change his strategy at this optimal point? In other words, does Elvis know bifurcations?
To find the optimal path, suppose that Elvis starts x1 meters out in the water and races to a ball that is z meters downshore and x2 meters out into the water. His swimming speed is s meters per second to a point y1 meters downshore; he then runs along the beach at speed r meters per second to a point y2 meters upshore from the ball, before finally swimming out to the ball.
Minimizing the total time for the swim-run-swim pathway reveals that, for the optimal path, the incoming swim leg makes the same angle with the shore as the outgoing swim leg does (like light reflecting from a mirror or a billiard ball bouncing off a rail). Pennings and Minton speculate that Elvis may somehow take advantage of this symmetry in determining the quickest path to the ball.
The mathematicians also worked out a formula for the bifurcation point at which the nature of the optimal solution switches from an all-swimming mode to the swim-run-swim strategy. Clearly, there's no bifurcation point if swimming is faster than running. For shorter distances, when running is faster than swimming, the small advantage that running provides can't compensate for the extra swimming distance to get to and from shore. As the ratio of running speed to swimming speed approaches infinity, the optimal path gets closer and closer to a trapezoid.
In the case of a ball thrown parallel to the shore, x2 = x1, and the bifurcation point is:
Because Elvis lives to fetch, it was possible to set up an experiment to test his bifurcation prowess. Pennings and two undergraduate students took Elvis to the beach and first conducted some time trials. They found that Elvis's running speed is about 6.39 meters per second and his swimming speed about 0.73 meter per second. However, when later chasing a ball, Elvis's running speed dropped to a leisurely 3.02 meters per second (it was, after all, a lazy July afternoon—and swimming is tiring). In this case, according to the mathematical model, the optimal bifurcation point is 2.56x.
The researchers conducted nine trials. Each time, Pennings stood 4 meters out in the water with Elvis and threw a ball various distances, but still 4 meters from the shore. One student measured the distance of the throw and the other recorded Elvis's choice.
Elvis was certainly smart enough to consistently take the swimming-only route for shorter distances and the swim-run-swim path for longer distances. For Elvis, the bifurcation point appeared to be somewhere between 14 and 15 meters. However, the formula predicts it should have been 10.24 meters. So Elvis missed the mark in three of the nine trials, choosing to swim rather than to take the longer but (in principle) quicker route.
"Thus," Pennings and Minton remark, "Elvis knows bifurcations qualitatively, but not quantitatively."
Minton, R., and T.J. Pennings. 2007. Do dogs know bifurcations? College Mathematics Journal 38(November):356-361.
Pennings, T.J. 2003. Do dogs know calculus? College Mathematics Journal 34(May):178-182.
Peterson, I. 2006. Calculating dogs. MAA Online (Feb. 20).
______. 2003. A dog, a ball, and calculus. MAA Online (June 9).