“π is wrong!” by Bob Palais
appeared in
The Mathematical Intelligencer Springer-Verlag New York Volume 23, Number 3, 2001, pp. 7-8.
The author gratefully acknowledges Dr. Chandler Davis, for his encouragement and editorial input.
(See also the Wikipedia entry on Dr. Davis.) The most amusing letter to the editor in response stated:
``I agree with Bob Palais' pi-ous article, but it may be 2-pi-ous.''
The article has been discussed in several places since then, including:
A couple of my own observations since the article. It seems to me that you can't have it both ways on area A= πr2 and circumference C= πd. If you believe diameter is fundamental, then it should be A= πd2/4. As noted in the last page of the pdf, I suggest calling the alternate constant 2π=6.283... `1 turn', so that 90 degrees is `a quarter turn', just as we would say in natural language. The main point is that the historical choice of the value of π obscures the benefit of radian measure. It is easy to see that 1/4 turn is more natural than 90° , but π/2 seems almost as arbitrary. It is apparent that we can't eliminate π but it seems helpful to be aware of its pitfalls, and introduce an alternative for those who might wish to use one.
A physicist quoted in the Shifty Constants column contended that these things don't matter as much as doing new science, and though in a narrow sense I agree in principle, in a broader sense I see a connection. Poor choices of the most fundamental constants deter future creative scientists, and hinder practicing ones. In physics, one of the few constants with the same impact as π is the speed of light, c. But what if c was defined as one-half the speed of light?! Every appearance of c would have to be changed to 2c, Lorentz transformations would involve (1-v2/(4c2))1/2, etc. I think the analogy is fair, and that such a situation would not promote the progress of physics! If someone tried to propose a remedy, they would undoubtedly face opposition, and claims that it makes no difference, but I would have to disagree.
Tau: τ
In 2010, momentum has grown for assigning the greek letter τ to represent the `circle constant' of `one turn', 6.283... Two `tau activists' leading the charge alerted me to their serious efforts to provide a consensus convention that would gain acceptance and wide usage, in the form of τ. The first, Dr. Peter Harremoes, is the editor of several respected scientific journals, lending significant credibility and visibility to the movement. He makes a strong case, and has found further precedent for the turn terminology (and decimal subdivisions!) in a definitive and classic 1962 book on Astronomy by the legendary astronomer, Fred Hoyle. He also found that as early as 1889, in his Traité D'Algebra, H. Laurent treated 2π as a single symbol, writing 2π/4 rather than π/2 . I was still cautious about not violating the design principle of non-conflict with existing uses, e.g. torque, time constants, and shear stress for τ, but began to seriously consider the proposal. Then, shortly before June 28 (6/28), I was also contacted by Michael Hartl to alert me to the impending release of the `Tau Manifesto'. My initial reaction to both emails was that no one needs my approval to make such a proposal, though at the same time it was very kind of them both to contact me for my involvement and support. The combined effect and independent arrival at the same conclusion by these two scientists forced me to rethink my opposition, and I eventually realized that though τ may conflict with previous use, it should not be difficult to avoid conflict in any future publication by choosing among the many reasonable alternatives whenever a torque or time constant is being discussed simultaneously. Therefore, I am pleased to lend my support to their case for τ and am honored to pass the torch that I truly inherited from predecessors like Hoyle to these and future advocates of more natural clarity in our notation for rotational measure!
Since then, I have noticed another potential conflict that is still (3/14/2011) not mentioned on the Wikipedia page on Tau, the torsion of a space curve that occurs in the Frenet-Serret equations. Amusingly, as that latter page is now written, it uses tau both for torsion and as a dummy variable of integration! This either shows the risks of ambiguous symbols, or perhaps it shows that they are not a big deal, and can be understood in context, even in close proximity!
The following \TeX macro which produces the symbol \newpi
used in the article was created by Richard Palais.
\def \newpi{{\pi\mskip -7.8 mu \pi}}