In this episode of My Favorite Theorem, Kevin Knudson and I were fortunate to have University of San Diego math professor Candice Price on the show. You can listen to the episode here or at kpknudson.com.

Dr. Price’s favorite theorem is John H. Conway’s basic theorem about the correspondence of rational tangles to the extended rational numbers. You are probably familiar with rational numbers: all the numbers that can be written as a ratio of two whole numbers. The “extended” bit just means we throw infinity into the pot as well. Rational tangles are knot-like objects that you can form by twisting pieces of string or rope in different ways. This video shows a group of students making some rational tangles. (For the slowed-down, sound on version, click here.)

The theorem is that each rational tangle can be paired uniquely with a rational number (or infinity), and likewise each rational number gets a rational tangle. So these complicated tangles can be boiled down to relatively familiar numbers.

The rational tangle dance is a popular activity for Math Circles and high school classrooms. Students can play with the twists and turns and try to figure out how they correspond to rational numbers. I especially love this video of some of Fawn Nguyen’s students finally figuring out how to untwist their tangle.

You can find suggestions for using rational tangles in classrooms or recreational math environments from Nguyen, Amie Albrecht, or Tom Davis (pdf).

Dr. Price first encountered rational tangles through Mariel Vazquez, who was a professor at San Francisco State University when Dr. Price started her master’s work there. These tangles show up in DNA topology, which both Dr. Vazquez and Dr. Price study. Yes, it turns out that these tangles resemble the way DNA folds up on itself! Dr. Price suggests the book Knot Theory and its Applications by Kunio Murasugi for more information on tangles, knots, and DNA.

Dr. Price decided to pair her theorem with a Neapolitan shake from In-N-Out Burger. You’ll have to listen to the episode to hear why it’s such a perfect accompaniment for Conway’s theorem about rational tangles.

This episode is airing during Black History month, and it just so happens that Dr. Price, along with Shelby Wilson, Raegan Higgins, and Erica Graham, runs a website called Mathematically Gifted and Black that highlights the work of contemporary black mathematicians during the month of February. I enjoyed following the site last year, and I’m glad it’s back again. The Notices of the American Mathematical Society also has an extensive section by and about black mathematicians this month (pdf).

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: