In this episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I were lucky enough to talk with James Tanton. Dr. Tanton is a former university mathematician and high school math teacher, but his current job title is “mathematician-at-large” of the Mathematical Association of America. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

As Dr. Tanton says in the episode, he views his role as one of creating and facilitating joyful, enriching experiences with mathematics for students at all levels of mathematics learning. Right now, he is especially excited about Global Math Week, a project that brings an activity called “Exploding Dots” to classrooms around the world during the week from October 10 to 17. Last year, he says, over a million students got to play with it in their classrooms; this year, they are aiming for 10 million.

My Favorite Theorem is about theorems, so of course we talked about one on the podcast. Dr. Tanton chose Sperner’s lemma, which is related to the Brouwer fixed-point theorem that both Francis Su and Holly Krieger talked about on their episodes of the podcast. Brouwer’s fixed-point theorem says that a function from a domain with no holes to itself must have a fixed point, a point that gets taken to itself by the function. Sperner’s lemma is sort of a discrete version of that continuous theorem.

Dr. Tanton explained how you can see the lemma for yourself. Draw a bunch of dots on a balloon, or flatten it out and draw a bunch of dots within a polygon on a piece of paper. Connect the dots into triangles in some way. Now label each point A, B, or C. No matter how you do this, on the balloon, if you find one triangle with all three letters represented on its vertices, there must be another triangle somewhere that also has all three letters. On the piece of paper, you need the additional requirement that there are an odd number of “AB” edges on the outside of the polygon. (That is, edges that connect a letter A to a letter B.) There are several different ways to state the lemma, so if you find other versions, you can have fun figuring out why they are equivalent.

Dr. Tanton also talked about how Sperner’s lemma can be used to prove the hairy ball theorem, famous largely because of its colorful name. This theorem says that on a hairy surface with no holes, like the outside of a tennis ball or coconut, there’s no way to comb the hair in a smooth way without at least one hair sticking straight up. The connection between the discrete nature of Sperner’s lemma, with its spread out dots, and the continuous hairy ball, was surprising to me.

In each episode of the podcast, we ask our guest to pair their theorem with something: food, beverage, art, music, or some other delight in life. Dr. Tanton paired Sperner’s lemma with his Aussie roots in the dessert pavlova. (Apologies to our Kiwi listeners; both Australia and New Zealand claim the pavlova, but Dr. Tanton is Australian.) You’ll have to listen to the episode to hear why this opulent meringue concoction is the perfect accompaniment to Sperner’s lemma.