Producer **Soren Wheeler** brings us a story about a friendship between Steve Strogatz and his high school math teacher, **Don Joffray**. Steve explains how numbers can connect you and where they fall short.

- A video trailer for Steve Strogatz's book "The Calculus of Friendship"
- Steve Strogatz's book "The Calculus of Friendship"

## Comments [13]

Thomas, Thanks for your question.

The idea is that a repeating wave can be described by a Fourier series. Roughly speaking, a Fourier series is made up of a "countably infinite" set of sine and cosine waves. Such an infinity is indexed by the *whole* numbers {0,1,2,3, ...}, and is the smallest kind of infinity.

In contrast, non-repeating waves can be described by a Fourier transform. A Fourier transform is made up of an "uncountably infinite" set of sine and cosine waves, which is essentially a continuous infinity of sines and cosines. This form of infinity is indexed by all the *real* numbers (not just the whole numbers) and is the higher form of infinity mentioned in the piece.

I would like to know more about the final comments on modeling ocean waves, waves that repeat versus waves that don't repeat, using different kinds of infinities. I've been reading about the infinity of real numbers between 1 and 2 and natural numbers (1,2,3,4,etc.) but I haven't been able to find how this applies to the study of the ocean waves. Can anyone point me in the right direction?

Had an NPR moment listening to this story! Just loved it!

What a treat to hear this wonderful account of teacher-'student- teacher. I had the pleasure of spending many summer afternoon's with Don Joffray. I lifeguarded at the beach in CT where Don would come to windsurf. I would bring my little brother for company and Don befriended us both. I happened to be taking calculus 2 at the time and Don would explain puzzling concepts to me with shells, sand, seaweed... I believe it might be the most pleasant way to learn Calulus, under the sun with the waves crashing in the background and an excellent teacher. My tutorials would be interrupted when the wind came up and Don would take to the sea on his board. The consummate teacher Don offered to teach my brother to windsurf. By the end of that perfect summer, I earned an A and Don and my little brother were windsurfing buddies. What a special, thoughtful, friendly, smart person. Thanks Don for your friendship.

Thank you for this great story

The story about the student and the math professor really touched me because I remember my first math-love-moment; I didn't know her name and she was gone the next day, but I wrote a poem to keep the feeling close.

An Ode to You, Mysterious Calculus Lady

For a brief second you lit my world

Your stare became y-axis of my rotation

Your glare made me quake with fear of non-real numbers

I cried out "Oh! Dear"

You were my Kajol*

It was love at first strike

when you wrote "the limit of the sine is one"

I understood and so did my heart

You became the integral to my derivative

Before you, I was discontinuous,

You were my Intermediate Value.

Between f(you) and g(me) was our future

I was the tangent to your curves

When you and I were simplified and factored,

we became one

Now you are gone

I am an irrational function with a denominator of zero

There is no one to conjugate my radical sadness

As X approaches Zero

I make no sense like Pi.

*Kajol is a famous Indian Bollywood Actress and is best remembered for her iconic role in the movie Dilwale Dulhania Le Jayenge (The Big Hearted Will Take the Bride).

While I agree that this was a nice segment, I don't think it belonged on this particular episode. Seems to me, it would have been better suited for inclusion on a program like "This American Life".

This is a beautiful segment. In so many ways my own experiences have paralleled the relationship revealed here, where passions for abstract concepts brought a closeness without being personable. It was inspiring to hear Steve's courage in moving past that invisible wall. I've only yet just begun in that process with many people I've known for quite some time, and find it encouraging to witness success in this.

It sounds as though Don was a penultimate teacher, understanding that in teaching one isn't always the knowledge carrier or the person who has all the right answers, but sometimes rather simply the one who just asks the right questions from a place of humility and humbleness. In that light, we all have the potential to be teachers.

Could someone post the name of the music used at the end of this story during the sound of the waves? It was perfect for that moment and I'd like to hear the rest of it, if possible - perhaps to contemplate this "greater infinity."

I don't want to pry, but, if possible, I would like to know how Mr. Joffray's oldest son Marshall died. My own brother died at 26, and I was immensely touched and reminded how events such as the death of a child/sibling are profound events, even many years later. I also know that people in my own family have difficulty talking about my brother's death because we never were ones for group grief experiences (WASPs to the core). I understand Prof. Strogatz's explanation for his own reluctance to address Marshall's death (a mathematician's approach of bifurcation), not because I am mathematically inclined (quite the opposite!) but because the human mind deals with unexpected tragedy in small, doable pieces of sadness.

Thank you for a lovely story by a teacher of a teacher.

Wow. You had me at 1. I'm still trying to wrap my head around: "25 Minutes to Go," Johnny Cash counts down the minutes to his hanging.

i want to now ho to love!!!!!!!!!!!!!!!!!!!

Hi Alex, It’s a good question. You’re right if you’re thinking that the outer border will be some kind of oval. But not all ovals are ellipses. An ellipse is a very specific kind of oval that satisfies certain algebraic equations, or, if you prefer, certain geometric conditions. The point is that if we consider a border of constant width around an elliptical swimming pool, its outer edge will be oval-shaped but will not satisfy the conditions needed to qualify as an ellipse. (If you like, I could send you a copy of my original letter to Mr. Joffray where I showed him a few proofs of this using calculus. Let me know; you can reach me at Cornell.) But if you just want an intuitive explanation, think about the limiting case where the elliptical pool is very long and narrow, like the shape of a cigar. Then if you put a one-foot border around it, the outer edge of that border will look like a football stadium, with two almost straight sides capped off by semicircles. That’s a very non-elliptical shape. The same difference holds true even if the pool is rounder and less cigar-shaped, but the proof is harder. Hope this helps…

Can someone please explain to me the elliptical swimming pool problem discussed in this segment. Steve says that the 1 foot border formed around the edge of an elliptical swimming pool will never be an ellipse. I would really like to know why or how this is the case.

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