On today’s episode of My Favorite Theorem, my cohost Kevin Knudson and I were happy to talk to Jayadev Athreya. He’s a math professor at the University of Washington in Seattle. He does research with my spouse, and we are happy to call him and his partner friends. (He mentions an amazing buttermilk pie I made for them one time, probably around 2012. The recipe is here. I skipped the sprinkle of sugar at the end and was not nearly as meticulous as the author when I made the crust.) You can listen to the episode either here or at kpknudson.com, which also has a transcript.

Appropriately for someone living in the Pacific Northwest, Dr. Athreya framed his favorite theorem in the language of trees. If you plant trees in a regular grid pattern and stand at the middle and look out, how many trees can you see? Of course, we need to add a few assumptions. Let’s assume you can see a total distance of R units and that a tree blocks any trees lying directly past it from your point of view, or lying in exactly the same direction from you. The problem can then be restated in less flowery (or leafy) terms as the number of points (m,k) in the plane that are less than R units away from the point (0,0) and where m and k share no common factors. (It's also called the Gauss circle problem.) The answer turns out to be 6/π2, a number that, along with its reciprocal π2/6, shows up many places in math.

For video introductions to similar tree visibility problems and the number π2/6, check out Numberphile and 3 Blue 1 Brown.

Dr. Athreya explained that this problem, which on its surface seems mostly related to number theory, also connects to geometry and dynamical systems, two of his main areas of research. By accident, this episode ended up being a sort of memorial for three remarkable mathematicians who passed away recently. Learn about Steve Mitchell from his autobiography (pdf) and the blog he wrote starting when he was diagnosed with bladder cancer and continuing until near the time of his death. Learn about William Veech from his Wikipedia page. And learn about Maryam Mirzakhani from the obituary Dr. Athreya wrote for the Indian math journal Bhāvanā, Anton Zorich’s article about her work from 2015 (pdf) after she became the first woman to be awarded the prestigious Fields Medal, or one of the articles collected here by the American Mathematical Society after her death.

In each episode of the podcast, we ask our guest to pair their theorem with food, beverage, art, music, or other delight in life. Dr. Athreya has both a wine and a beer pairing for his favorite theorem. He chose primitivo wine, which as he notes could have been served at the Last Supper, or a spontaneously fermented beer like a lambic. Whether you’re an oenophile or zythologist (which is a real word, red squiggles be darned), he has you covered. You’ll have to listen to the episode to find out why those are perfect accompaniments to lattices, primitive points, and the ubiquitous number 6/π2.

In addition to teaching and research, Dr. Athreya is the director of the Washington Experimental Math Lab, which is one of several geometry labs in universities around the country that do student research and training in addition to math outreach and public engagement. Find out more at their website, WXML, not to be confused with the Christian radio station in Ohio.

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: