On this episode of My Favorite Theorem, Kevin Knudson and I talked with Dave Richeson. He’s a math professor at Dickinson College in Carlisle, Pennsylvania and editor of Math Horizons, a magazine published by the Mathematical Association of America. You can listen here or at kpknudson.com, which also has a transcript of the episode.


Richeson has been on a bit of a Greek mathematics kick, so he chose to tell us about Archimedes’ theorem that π is a constant. That is, the ratio of a circle’s circumference to its diameter is the same, no matter the circle. (Richeson argues that we should be calling it Archimedes' constant instead of π.) As he mentions, it’s not well-known among mathematicians who first showed that it was a constant, much less determined what that constant was. Richeson said mathematicians often guess it was Euclid, but that Euclid’s axioms and postulates aren’t quite enough to prove it. Not only did Archimedes (287-212 BCE) show that the ratio was always the same, he also figured out that it's between 223/71 and 22/7.

Archimedes compared the perimeters of inscribed (left) and circumscribed (right) polygons to come up with his estimates for the number we now call π. He used 96-gons to get his most accurate estimates, but the method is more clear with hexagons because 96-gons look too much like circles. Credit: Urutseg Wikimedia

Richeson wrote about the history of this theorem in the College Math Journal. (A version of the article is also available for free without a subscription on the preprint server arxiv.org.) He says at first he was embarrassed to ask who proved that a circle's circumference:diameter was a constant, so one of the morals of the episode is that it's OK to ask those slightly embarrassing questions. They might take you on interesting adventures!

On the episode, we talked a little bit about the history and mythology surrounding Archimedes. Mythbusters even did an episode about the story that he used large mirrors to set enemy ships on fire. You can read about that story on the blog Oddly Historical as well. The story of his death was recorded by Roman historian Livy and others. To add a story to a story, the story is that the story of Archimedes' death is one of the things that got French mathematician Sophie Germain interested in studying math.

A nice, hot pizza is the perfect food to enjoy while pondering Archimedes and circle constants. Credit: 4028mdk09 Wikimedia (CC BY-SA 3.0)

What's the best pairing with a theorem all about circles? Pizza! Pizza is definitely the best food (according to me), so I heartily endorse that choice. You might be able to guess why Richeson chose that pairing, but you’ll have to listen to the episode to find out which toppings he would choose to enjoy on his pie. 

You can find Richeson online on Twitter (he’s @divbyzero) or at his blog Division by Zero, where he recently wrote a neat post about the mathematics of a reflected double rainbow. His book Euler’s Gem about a completely different (but also great) theorem was published by Princeton University Press in 2008.

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 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:

Episode 0: Your hosts' favorite theorems
Episode 1: Amie Wilkinson’s favorite theorem