## New Lives for Old Problems

December 4, 2009

Book Review
Gilbert Strang

The Calculus of Friendship. By Steven Strogatz, Princeton University Press, Princeton, New Jersey, 166 pages, \$19.95.

I confess to being an autobiographoholic. Good authors with interesting lives (their own) are irresistible. In the local library, I go to the third shelf and pay \$1.50 to read about someone's life (mostly I skip Hollywood). The librarians must wonder why this guy can't live his own life.

Right now the book is Rocket Boys by Homer Hickam. Parts of it are sensationally good. I tell you this so you will know what else I am thinking about as I read and review The Calculus of Friendship.

This book really has two authors---both Steve Strogatz. One of them, the mature one at Cornell with an excellent text on chaos, writes beautifully. As you read the first pages of this book, you smile at such good sentences. Those pages set the stage for his correspondence with his high school calculus teacher, Mr. Joffray. Their letters are the heart of the book, and the reason for its title, and the difficult part to review.

The letters are about specific calculus problems, most of them familiar. By happy chance, several have new lives. When four dogs chase each other cyclically, starting from the corners of a unit square, the key point is the distance before they meet (it is 1). In a short talk in Denver,* Nick Trefethen moved the problem to a rectangle, and computed the astonishing speed of spin and final collapse (nontrivial asymptotics).

For triangles, Hero (or Heron) found the area from the lengths of the sides. After growing up, I never expected to see that formula again. But it appears at the end of this book, with a proof. And on the blackboard in my own office, Alan Edelman has used that formula in his picture of triangle space. Triangles are points on a hemisphere, and Hero's formula connects their areas to their heights above the plane of the equator.

This picture answers a question that Lewis Carroll asked in 1858: Are random triangles acute or obtuse? Short answer: Three-quarters of that hemisphere contains obtuse triangles. And there is another odd connection to Lewis Carroll. The "Monty Hall problem" of choosing among three doors also comes up in this book. Steve and Mr. Joffray discuss Marilyn vos Savant and her correct strategy. In his list of Pillow Problems, Lewis Carroll asked and answered the same question!

The transition from student and teacher to the reverse comes pretty early. Steve remarks that their personal lives don't enter the letters very strongly: This is not Tuesdays with Morrie---the letters are truly about calculus.

It is hard to know how SIAM readers will react. Do we still care about problems we once practiced on? I guess they helped teach us to think in a more or less straight line. Maybe we are not sentimental about learning calculus, but it changed our lives.

Gilbert Strang is a professor of mathematics at MIT.

*"Four Bugs on a Rectangle," 2009 SIAM Annual Meeting.

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