Some readers may be familiar with Bob Palais’ article “π Is Wrong”. Within it Palais argues that π is the wrong choice of circle constant. This quote, from the author’s website, summarizes his main argument:
As noted in the last page of the pdf, I suggest calling the alternate constant 2 π=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 is to be aware of its pitfalls, and introduce an alternative for those who might wish to use one.
— Bob Palais
Palais then goes on to define a “newpi” symbol through a TeX macro, which resembles the fusion of two π:
The aforementioned article has been in print since 2001, and very little has changed on this front since then. The ideas it put forth are an amusing opinion that many of us tend to agree with, but 2π has not been adopted by the mathematical community.
Today Michael Hartl announced “The Tau Manifesto” on what he calls Tau Day (6/28 for 6.28…). In this document, Hartl echoes the good points that Palais made and builds upon them to construct a strong case in favor of adopting a circle constant which is the ratio of a circle’s circumference to its radius, not its diameter. Inspired by Palais’ use of the word “turn”, he proposes calling this constant τ (tau).
As Hartl argues, this symbol already exists (unlike the odd symbol that Palais introduced), it’s still generally available in mathematics, and it resembles π.
This new constant would not only be an improvement from a pedagogical standpoint (as shown in the diagram above), but would also “improve” several formulas. For example, Euler’s identity:
Which would become neater as:
This makes sense intuitively (a rotation in the complex plane by one turn is 1).
(The Tau Manifesto addresses the issue of how this too can relate to the “five most important numbers in mathematics” with a slight rearrangement.)
What are your thoughts on this? As mathematics evolves, is it time to start using “Let τ = 2π” as a means of adopting a better circle constant?
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