Forget Pi, here comes Tau
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:
Or:
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?
I like the idea but I really doubt it’ll ever become mainstream.
I have conducted a personnal survey: almost nobody in the real life knows what pi is, even if they always know its value by heart. So tau has its chance in the mainstream…
Pi is there because it is the ratio between the diameter of a circle and it’s circumference. No one measures the radius of a circle, it makes much more sense to just measure how “wide” a circle is.
Jordan, then why do we have
and 
? 
I’m saying physically measuring it. That’s why it’s C = pi * d
Measuring the radius is harder than measuring a diameter. And the area formula works out better with pi and not this tau thing.
Actually, 1/2 tau r^2 is more “natural” than the current formula. It has more in common with familiar formulas such as 1/2 m v^2 etc.
Hardly. 1/2 tau r^2 makes it clear that you’ve got a triangle. There’s always a 1/2 with triangles.
that would make for a symbollic type of relationship seeing as if you draw circle inside of a triangle inside of a circle (where the perimeters of each inner shape symmetrically intersect 3 times) the relationship between the 2 circles is a 2:1 diameter.
Tau is a bad symbol for this. Two situations where tau is used, torque and time constants, make frequent use of pi.
Also shearing stress is tau…might not be the best symbol used?
my god!!
THEN TODAY IS TAU DAY!! (6/28) !!
The way I learned about radians back in PreCalculus was that 1 radian was the angle on the unit-circle such that the arc along the circle across that angle is of length 1.
I didn’t (and still don’t) think of radians in terms of PI. I think of radians in terms of radii and arcs. PI is a nice convenient fact to remember about the radii and arcs.
This would also make many physics equations more intuitive. There are many times where the constant 2(pi) appears, and far fewer where (pi) is by itself.
I’m all for it. Pragmatism++
My colleague Andrej Cherkaev likes to point out that rather than an inscribed triangle, an inscribed hexagon (or 6 equilateral triangles of side 1) gives a very natural estimate for the circumference being close to, and slightly greater than 6 times the radius. ( Andrej and Elena have a very fun Math Jokes page:
http://www.math.utah.edu/~cherk/mathjokes.html
as well as nice pages on their interesting work on non-convex variational problems and composite materials:
http://www.math.utah.edu/~cherk/mathmat/index.html
http://www.math.utah.edu/~elena/ )
If it’s truly useful it will be adopted. The wonderful thing about mathematical reasoning is that there is nothing inherently wrong with setting tau = 2*pi and using some symbol for tau. If it helps convey reasoning, there is no mathematical precedent which forbids you setting the above relationship. Whether or not it is more ‘organic’ in terms of teaching math, I do not know. I think any apt math pupil would not be thrown by either interpretation, and there is no reason to redefine trig functions in terms of tau under the assumption that it would help the average student who only cares for the bare minimum of a passing grade in math.
Also, just would like to add: defining a constant tau=2*pi in no way ‘does away with’ the irrational constant pi, for tau is a multiple of pi. We strive to get at the most basic building blocks of geometric shapes, and pi is there whether we like it or not. This really seems to be an aesthetic issue/argument, IMO.
hello, greetings to all …
my name is jose I’m Chilean and student teachers in mathematics, surfing the net I found this blog and my attention this post. I find great idea, this would bring super positive consequences for teachers of mathematics, since it is my hard to understand why many students circle the circumference is 2 pi, after which they are taught that the perimeter of a circle is 2 pi plus radio, this brings many Confucianism for students and makes us very difficult to make classes.
I am in favor of the change from pi to tau.