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Read the State of the Tau 2026  | 

State of the Tau 2026

Happy Tau Day, everyone!

When I first proposed using the Greek letter $\tau$ (tau) for the circle constant in 2010, I never imagined it would still be going so strong 16 years later. But here we are on 2026-06-28, and the number $\tau = C/r = 6.28\ldots$ remains as relevant as ever. I’d especially like to express my gratitude to you, since Tau Day would be nothing without all the terrific tauists who’ve made it all the way to tau’s Sweet Sixteen.

And of course it’s twice as sweet with twice as much pie! 😋

Tau’s Sweet Sixteen

One of the most inspiring things about Tau Day is the sheer variety of different ways people celebrate it. In honor of tau’s Sweet Sixteen, here are 16 examples from last year:

Among many standouts, I’m especially pleased to see pioneering quantum information theorist John Preskill on the list above. (Technically, Preskill’s post was on Half Tau Day, but I’m going to count it.) Preskill probably wouldn’t remember this, but I actually had him for a physics course (Physics 136c) back when I was a graduate student at Caltech. He was also the Ph.D. advisor for my longtime friend Sumit Daftuar, who gets a separate shout-out in the acknowledgments of The Tau Manifesto, second only to “$\pi$ Is Wrong!” author Bob Palais.

Tau-pilling Claude, et al.

Speaking of Bob Palais, Bob sent me this wonderful conversation he had with the Claude AI chatbot about pi and tau, which I share here with his permission. Although Claude initially shows a “soft spot for π (pi)”, with Bob’s guidance it quickly agrees that the “pedagogical costs [of pi] are real”, adding that

With τ, the unit circle just works — a quarter turn is τ/4, a half turn is τ/2. The fractions mean what they say.

You can read the whole thing for a delightful window into the current state of the art in large language models, viewed through the lens of pi vs. tau.

In addition to the Claude conversation, Bob passed along a video a colleague sent him, which has an excellent discussion of the volume of an $n$-dimensional sphere. Created by the inimitable 3Blue1Brown (who has weighed in on tau before), the video includes a discussion of how the volume of an $n$-sphere varies with dimension $n$. A glance at a screenshot of the talk shows some rather prominent factors of $2\pi$:

3Blue1Brown on the volume of a sphere

Bob also sent a link to the surprisingly recent Scientific American article “Why some mathematicians think we should abandon pi” by theoretical physicist Manon Bischoff. Originally published in the German-language version of the magazine (Spektrum der Wissenschaft), the article includes an excellent history of tau and aptly summarizes the relevant arguments. It stands as a nice bookend to “The Tao of Tau” by Elizabeth Landau, published on the Scientific American blog in 2017, which was one of the earlier mainstream articles that took tau (somewhat) seriously.

The Fall of Rome

I was particularly tickled by the following post, which presumably ties into the “How Often Do You Think About the Roman Empire?” meme by offering a novel theory on how Rome fell.

A gem from Ramanujan

Finally, I’d like to share this remarkable continued fraction formula I came across in the book Elementary Number Theory by David Burton (p. 320):

\[e^{2\pi/5} \left( \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{1 + \sqrt{5}}{2} \right) = \frac{1}{1 + \dfrac{e^{-2\pi}}{1 + \dfrac{e^{-4\pi}}{1 + \dfrac{e^{-6\pi}}{1 + \cdots}}}}\]

Experienced math nerds may recognize the fingerprints of Srinivasa Ramanujan, and indeed Ramanujan included the above formula in a 1913 letter he sent to distinguished British mathematician G. H. Hardy, leading to a famous collaboration between the two. Ramanujan’s ebullient genius produced a seemingly endless variety of similarly incredible formulas, but I’d like to humbly suggest one small edit in this case:

\[e^{\tau/5} \left( \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{1 + \sqrt{5}}{2} \right) = \frac{1}{1 + \dfrac{e^{-\tau}}{1 + \dfrac{e^{-2\tau}}{1 + \dfrac{e^{-3\tau}}{1 + \cdots}}}}\]

We’ll never know if Ramanujan might have agreed that the formula is better with tau. One thing is for certain, though: 2026-06-28 is a great date to celebrate the circle constant at any rate!

Michael Hartl
Founder, Tau Day
Author, The Tau Manifesto

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