Posted on 02/18/2006 2:42:48 PM PST by T-Bird45
It all started with Elvis.
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.
When Elvis and Pennings go to the beach, they always play fetch. Standing at the water's edge, Pennings throws a tennis ball out into the waves, and Elvis eagerly retrieves it. When Pennings throws the ball at an angle to the shoreline, Elvis has several options. He can run along the beach until he is directly opposite the ball, then swim out to get it. Or he can plunge into the water right away and swim all the way to the ball. What happens most the time, however, is that Elvis runs part of the way along the beach, then swims out to the ball.
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 beach—in 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.
Paths to the ball. Shoreline distance AC = z; perpendicular distance to target BC = x; DC = y; AB = w.
Here's their argument. When running from A towards C, the ball at B appears closer and closer as the dog gets closer to C, but its speed of approach to B diminishes (reaching zero at C). At some moment of its run, its speed of approach while running on the beach equals its speed of approach when swimming directly to the ball. If the dog jumps into the water at this moment, the strategy yields the same y value as that provided by the travel-time minimization model (where r is the dog's running speed, and s is its swimming speed).
"Although this solution is identical to that proposed by Pennings," Perruchet and Gallego say, "it was gained without assuming canine knowledge of the entire route, and hence can be construed as a more plausible model for [the] dog's strategy."
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).
My Jack Russells are pretty smart sometimes.
Yup.. and the boys of summer make physics calculations while standing in the batter's box.
They do calculus and urinate on the rug. They are geniuses all right..
Ping!
But it's OK as long as they take the optimal path to do so, right?
Yep, this is fabulous science. I would have never known that dogs know how to fetch a thrown ball with accuracy. I'll bet flying birds use thermals with a precision similar to mathematical models showing how to best soar through the air; dolphins cut through the water in a manner reducable to numbers that demonstrate that they use the least amount of muscular power for the amount of forward motion; and wolves run down prey reflecting a near-perfect relationship between energy used to maintain the chase and finally killing the quarry. Wow.
I suspect it's the same with the dog. The fallacy in the writer's theory is that calculus came first and the behavior follows. In truth, calculus is simply a tool used to describe the behavior that already existed.
Very funny topic, I must say. I wouldn't be at all surprised if there is, indeed, some mathematical process going on in the dog's brain. But then, we'll never know for sure, will we? If I could venture a guess, I would have to say that it is some sort of successive approximation process.
duh, the idiot has it the wrong way around...the mathematical formula closely matches the optimal way the dog, Elvis performs this particular task!
The dog is doing this intuitively. There is no "dV/dT" calculation going on, the dog is simply estimating (based on experience) where the ball (or Frisby, or stick) is going to be at a particular time, and speeds up, slows, or turns to meet the object at that particular point. As the dog has its eye on the object, it continues to adjust the estimate to meet the anticipated contact point at the optimal time.
Human beings do the same thing when trying to beat out a red light.
I hate it when my "bread" doesn't integrate. Keeps me up for hours... sometimes with gas.
How do you apply the statistical variable on the probability of a cop being near enough to observe the results and act accordingly? :-)
I can catagorically state that there is NO process going on in my dog's brain. He's like a male, human teenager.... sleep, eat, defecate (he doesn't think about the Other Subject, he's been repaired).
The catz, on the other hand, try to do math. Most times, they're pretty good. Integration, summation, correlation.... but they sometimes fail spectacularly. The other morning, one of them pulled all of the previous night's washed dishes down on her head when she snagged the bar-mop (dish towel) with a claw.
And sometimes, they calculate the amount they roll around asleep incorrectly, and roll off of the ironing board, landing wide awake, and looking around to see who saw...
A dog doesn't know enough to care who saw. He doesn't attempt mathmatics.
8>)
/johnny
They told me blue heelers were smart, but mine is almost 4 and barely reads.
One shouldn't ascribe to mathematics that which instinct and common sense can achieve in 1/10000 of the time.
Must be a different kind of calculus than my dog used; the one who did not know how to use the doggie door until after I got on my hands and knees and modeled it for him a few times and finally pushed him through it.
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