Theorem of the Day

Theorem of the Day and the τ Manifesto

In a 2001 article in Mathematical Intelligencer, Bob Palais argued persuasively that the domination of circle geometry by a constant representing half a circle was perverse, obfuscatory and pedagogically unsound.

Michael Hartl was inspired to ask whether this domination might not be overthrown. His Tau Manifesto, launched in conjunction with a substitute for the internationally recognised Pi Day, maintained the light touch of Palais but assembled a weighty 'case for the prosecution'. Certainly there seems to be a case for Pi to answer; and its main (although certainly not only) defence seems to be that possession is 9/10 of the law.

theoremoftheday is happy to do its bit to promote debate, and that is best done by taking a stand. Accordingly, in all theorem descriptions posted here which involve a circle constant, this constant has been expressed in terms of τ = 6.2831... rather than π = 3.1415.... Such theorems are marked with the above logo (home-grown — it is not official!)

Some remarks:

• the issue of optimising mathematical notation is recurring and enormously varied. I have collected a few examples here. Valuable historical context is provided by Jeff Miller here.
• the number of theorems here whose actual statements feature a circle constant is very small; for working mathematicians π vs τ is not a big deal!
• discussion of whether certain important formulae or theorems are more 'natural' or 'elegant' one way or the other threatens to be counterproductive: Girard's Theorem is shorter with π; Stirling's approximation with τ; mostly (Kepler's Conjecture, say) it seems presumptuous to pretend we know whether either is 'right'. This blog entry, although not without interest, is an extreme example.
• nevertheless, replacing a clear-cut π with a τ/2 in the context of circle geometry (Girard's Theorem is a case in point) should not make you say "ugly!" it should make you say "oh, why is a half-circle involved here?". Such questions are often worth asking!
• and where circles are involved, τ does generally seem more suggestive. If the identity e^iτ = 1 amounts to a definition of the unit circle, then e^iτ/2 = –1 has to be the way to describe a half-circle (and it seems inadequate to dismiss this as 'notational': Euler's Identity is the climax of a linguistic journey which begins with a mere notational convenience: a²= a×a, but there is so much more to mathematical notation than convenient syntax).

Here are some more links. If you have written or posted anything on this subject please feel free to offer a link.

• What Wikipedia (English language) currently offers on the subject; there is more at SimpleWiki
Joseph Lindenberg's Tau Before it was Cool
• Al-Kashi’s constant τ by Peter Harremoës (includes an extensive list of further Tau-links)
• A Pi Manifesto
• My Conversion to Tauism by Stephen Abbott
• Some reflections by Kevin Houston
• Atomic Spin on Defending Tau
• Aperiodical hosts a Pi vs Tau debate
• and the same protagonists do it on film for numberphile
• Tau at the Khan Academy
• Oxford University hosts a day-school on Tau vs Pi
• Some text-book quality notes by Ben Hummon explaining and using Tau
• The highly reputable plus magazine declares in favour of Tau...
• ... and includes this very persuasive clip by Phil Moriaty
• The classiest Tau diagram I've seen so far: hexnet.org/content/dozenal-tau-unit-circle; so classy that cadence have turned it into a watch!
• Tau features in regular cartoon strips SMBC and xkcd!
• And on page 90 of a classy AMS-published book on Axiomatic Geometry
• John Horton Conway supports Tau (free to read once you've registered)
• Art Benjamin supports Tau (see Twitter Q&A following interview)
• Top maths blog Annoying Precision declares in favour of Tau
• Scientific American carries an article in favour of Tau
• Top blog prooffreader.com finds 'randomesque' reasons for preferring Tau, and offers a one billion digit Tau download!
• I you just need 10⁵ digits they are here.
• "Pieces of Centaurs" a constrained-writing Tau-based poem by Mike Keith.
• 2014 saw a good deal of blogging about Tau around the 28th June. There is a round up in Carnival of Mathematics 112.
• Tangential but of great interest is the copyright furore over π, which might potentially replicate for any mathematical constant!
• A short (<15mins) recent (2015) presentation of the case for Tau is given by Michael Hartl here.
• And on 'super pi day' (3/14/15) Vi Hart complains eloquently about the arbitrariness of date-based jamborees (and about pi as a circle constant).
• While Andrea Hawksley suggests instead a 'non-demoninational' celebration of circles on the 20th March equinox
• 2π is referred to (punningly) as 'Planck's constant' by Jonathan King in his definition of a natural measure for solving Tarski's Plank Problem, in "Three Problems in Search of a Measure", The American Mathematics Monthly, Vol. 101 (1994), pp. 609–628, online here
• The website taufortrig.org offers a range of links and resources for those aspiring to pi-free teaching
• Thomas Cool has made many contributions to the debate on definition and nomenclature in trigonometry and in other parts of mathematics. An entry point is here.