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'. 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 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.