Do dogs know calculus?

What is calculus? Calculus is the study of change, in the same way that geometry is the study of shape. It is part of modern mathematics education.

Do dogs know calculus?
In 2003, Tim Pennings (an associate professor at Hope College) performed an experiment with his Corgi named Elvis

The experiment
He took Elvis to the beach and threw a ball into the water


What path did Elvis take?
One possible path was for Elvis to simply jump into the water and swim directly to the ball (illustrated by the red line) Such a path minimised distance to the ball.

But if Elvis didn’t like the water, he might have sprinted down the beach to the point on shore closest to the ball and then turned at a right angle and swim to the ball (illustrated by the blue line)

Surprisingly, Elvis did neither. Instead, Elvis ran part of the way along the beach, and then at some mystery point he plunged into the water swimming diagonally to the ball (illustrated by the green line)

Why did Elvis take this path?

  • Curiously, Elvis took the path that minimised the time it took to reach the ball!
  • By using calculus, Prof Pennings was able to show that Elvis almost always took a path that was close to the optimal solution (he computed the optimal path after performing various measurements of distances and speed).
  • Prof Pennings published his findings in a (now popular) research paper: Do Dogs Know Calculus? TJ Pennings, The College Mathematics Journal, Volume 34, No 3 (May, 2003), pp 178-182.
  • His discovery has received international media attention.
  • CNN clip on YouTube:

Mathematician TJ Pennings has written a paper on his dog Elvis’s use of calculus. He discovered that if he threw a ball into the sea, Elvis was able to calculate the shortest route to retrieve it. Taking into account the various required angles and distances of running on the beach sand (quicker) or swimming through the water (slower), and at which point these variables were most efficiently utilised, Elvis could expedite retrieval. The dog’s choice of route was confirmed by calculus-based maths to be consistently the most expedient.


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