Happy birthday to us! Kevin Knudson and I published our first episodes of My Favorite theorem a year ago. You can experience the nostalgia with us by listening to our first episodes, Episode 0 and Episode 1. We were thrilled to have Ingrid Daubechies as our guest for our anniversary episode. She is a professor at Duke University, recipient of a MacArthur Foundation fellowship, and past president of the International Mathematical Union. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

Dr. Daubechies had recently learned about Tutte’s embedding theorem, or Tutte’s spring theorem, so she chose to talk about it for the podcast. This is a theorem from graph theory, which means it’s about collections of points with edges connecting them, like a subway map or the social connections on a social networking site. The theorem states that graphs satisfying certain conditions can be drawn very nicely inside of convex polygons in the plane. (A convex polygon is one with no sticky-inny bits. More formally, the line connecting any two points in the polygon remains inside the polygon.) Furthermore, the theorem gives you an algorithm for finding this nice embedding of the graph into the plane. Dr. Daubechies described it as using a hair dryer on Saran wrap and having it shrink into a tight conformation.

The conditions necessary for a graph to have a Tutte embedding get a little technical, but one type of graph that works is a mesh, like a square grid or a triangulation. That makes Tutte’s theorem useful in many areas of applied mathematics, especially graphics. Dr. Daubechies was interested in the theorem because of a mathematical biology project she is working on. She and her collaborators—some mathematicians and some biologists—are using mathematics to study tooth evolution. The graphs they are interested in come from modeling the surface of a tooth as a triangulated graph. They then work on ways to quantify the differences between teeth of different animals. You can listen to her talk about this project in this video.

We also talked about her approach to collaboration and her work using math to help art historians with conservation and restoration. She wrote about that work for Quanta. In each episode, we ask our guest to pair their theorem with something. Dr. Daubechies chose to go with handicrafts: crochet, knitting, and sequins. You’ll have to listen to the episode to hear why they are the perfect accompaniments for Tutte’s embedding theorem.

Kevin and I are delighted to have brought you a year of favorite theorems from mathematicians around the world and across many different mathematical fields and careers. If you're a fan, we’d love for you to share your favorite Favorite Theorem with a friend who might enjoy it. Thanks for listening!

You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at kpknudson.com and here at Roots of Unity. A transcript is available here. You can subscribe to and review the podcast on iTunes and other podcast delivery systems. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Kevin Knudson’s handle on Twitter is @niveknosdunk, and mine is @evelynjlamb. The show itself also has a Twitter feed: @myfavethm and a Facebook page. Join us next time to learn another fascinating piece of mathematics.

Previously on My Favorite Theorem: