Theorem of the Day

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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 = 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).

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