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!)
- 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'.
- 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 eiτ = 1 amounts to a definition of the unit circle, then eiτ/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).
- anyway, circles are so fundamental to mathematics, and a change of notation is so easy to accomplish (this website has achieved it in a matter of hours!)...
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.