This article is dedicated to defend one of the most important numbers in mathematics: $\pi$.
Quite recently, a phenomenon known as the *Tau Movement* has steadily grown and is gaining more and more followers (called *Tauists*) by the day.
This is largely due to three driving forces:

- The original article
*$\pi$ is wrong*written by Bob Palais (published in 2000/2001). - The Tau Manifesto written by Michael Hartl (launched on June 28th, 2010).
- The video Pi is (still) wrong by Vi Hart (uploaded on March 14th, 2011).

Tauists claim that $\pi$ is the wrong circle constant and believe the true circle constant should be $\tau=2\pi$. They celebrate Tau Day (June 28th), wear $\tau$-shirts and spread pro-tau propoganda.

In this article we will explore this very question and provide several reasons why $\pi$ will prevail in the intriguing $\pi$ versus $\tau$ battle.But are tauists doing more harm than good?

The buzz around the blogosphere and on various online news sites is that there is a battle happening in mathematics, namely $\pi$ versus $\tau$.
Headlines in newspapers and on blog articles often declare that *$\pi$ is wrong* and tend to mislead the general public:

- Mathematicians want pi out tau in (SundayTimes.lk)
- Down with ugly pi, long live elegant Tau, physicist urges (TheStar.com)
- Mathematicians want to say goodbye to pi (LiveScience.com)
- On national tau day, pi under attack (FoxNews.com)

According to an article published by The Telegraph on Tau day:

"Leading mathematicians in India, the UK and the US appeared oblivious to this campaign today and asserted that there has been no debate or even discussion over replacing $2\pi$ with $\tau$ in serious mathematical circles."Mathematician Alexandru Ionescu at Princeton University says:

"Either one is just fine, it won't make any difference to mathematics."Siddhartha Gadgil, a mathematician at the IISc, says:

"The whole notion of replacing $\pi$ by $2\pi$ is silly since we all are very comfortable with $\pi$ and multiplication by two."

In fact, one grad student in mathematics goes on to say:

"Of course it had to be a physicist who would want to get rid of the usage of $\pi$... Theconceptof $\pi$ has been around since the time of the ancient Babylonians (the greek letter representing this number was popularized by Euler in the 18th century)... so why change now and trash it? This isn't the first thing that physicists have tried to change in the field of mathematics (notation wise, anyways). I for one believe that the mathematics community will not be lemmings here and go with this idea; I know I'm certainly not going to accept tau as a replacement for pi."

It is debatable whether the media coverage of $\tau$ is good publicity or bad publicity for mathematics, but regardless, the Tau Movement has definitely sparked an interest. Even those with very little mathematical background are curious about it! I think most mathematicians would agree that anything that generates interest in math is a definite plus.

As seen from the quotes above, a lot of mathematicians simply shrug off the Tau Movement as being silly. In this article we attempt to give a serious rebuttal to $\tau$ in the defence of $\pi$. Any suggestions and reasons why $\pi$ is better than $\tau$ (or $\tau$ is better than $\pi$) are more than welcome!

Tauists argue that by using the constant $\tau=2\pi$ a lot of formulas become simpler. Unfortunately, the Tao Manifesto is full of selective bias in order to convince readers of the benefits of $\tau$ over $\pi$. They pinpoint formulas that contain $2\pi$ while ignoring other formulas that do not. We demonstrate below that when making the change to $\tau$, there are lots of formulas that either become worse or have no clear advantage of using $\tau$ over $\pi$. Tauists also claim that their version of Euler's formula is better than the original, but we will see that it is in fact weaker. The benefits of $\tau$ only appear when viewing $\pi$ from a narrow minded two dimensional geometrical point of view, but these benefits disappear when looking at the bigger picture. We will see how the importance of $\pi$ shines through as it shows up all over mathematics and not just in elementary geometry.

The Tau Manifesto relies on the traditional definition of $\pi$, namely, the constant that is equal to the ratio of a circle's circumference to its diameter: $$\pi\equiv\frac{C}{D}\approx 3.14159\ldots.$$ The manifesto then goes on to suggest that we should be more focused on the ratio of a circle's circumference to its radius: $$\tau\equiv\frac{C}{r}\approx 6.283185\ldots.$$ In particular, since a circle is defined as the set of points a fixed distance (i.e., the radius) from a given point, a more natural definition for the circle constant uses $r$ in place of $D$.

So why did mathematicians define it using the diameter? Likely because it is easier to measure the diameter of a circular object than it is to measure its radius. In the Tau Manifesto, Hartl says:

"I’m surprised that Archimedes, who famously approximated the circle constant, didn't realize that $C/r$ is the more fundamental number. I’m even more surprised that Euler didn't correct the problem when he had the chance."But Dr. Hartl, there is no problem to correct, $\pi$ is not wrong, and we will soon see that we have been using the right constant all along.

There are numerous reasons to define the circle constant using $\frac{C}{D}$. Some of these reasons include:

- This definition is consistent with the area definition discussed in the next section.
- In practice, the only way to measure the radius of a circle is to first measure the diameter and divide by $2$.
- Why look at a ratio where you go all the way around the circle yet only HALF way across it? It just doesn't seem natural.
- Some believe the Bible says we should be looking at circumference and diameter, not the radius. (Author's note: This isn't a serious reason :P)

Another definition for $\pi$ is to define it to be twice the smallest positive $x$ for which $\cos(x)=0$ [4], or the smallest positive $x$ for which $\sin(x)=0$. With this definition neither $\pi$ nor $\tau$ is simpler than the other. Tauists may claim that $\tau$ can be defined as the period of $\cos(x)$ or $\sin(x)$ but whether this is better is up for debate (in the same way, $\pi$ can be defined as the period of $\tan(x)$).

Another common geometric definition for $\pi$ is in terms of areas rather than lengths. Take $r$ to be the radius of a circle.
Define $\pi$ to be the ratio of the circle's area to the area of a square whose side length is equal to $r$, that is,
$$\pi\equiv\frac{A}{r^2}.$$
In terms of $\tau$, this definition is messy and includes a factor of $2$.
In particular, define $\tau$ to be the twice the ratio of a circle's area to the area of a square whose side length is equal to $r$, that is,
$$\tau\equiv 2\left(\frac{A}{r^2}\right).$$
Clearly, this definition favors $\pi$ over $\tau$ and also involves the *important* radius of a circle.
Like the traditional definition, this definition of $\pi$ depends on results of Euclidean geometry and comes naturally when looking at areas.

Mixing things up a bit, in terms of diameter we can define a constant (call it $\pi/4$) as follows: $$\frac{\pi}{4}\equiv\frac{A}{D^2}.$$ This suggests that perhaps both $\pi$ and $\tau$ are wrong, and $\pi/4$ is the correct circle constant. Others have also suggested similar numbers as the circle constant. In 1958, Eagle suggests that $\pi/2$ is the correct circle constant [1]. In fact, the $\pi/2$ Manifesto is coming soon to a website near you! (Just kidding, I hope). But why stop at redefining $\pi$? Terry Tao says:

"It may be that $2\pi i$ is an even more fundamental constant than $2\pi$ or $\pi$. It is, after all, the generator of $\log(1)$. The fact that so many formulae involving $\pi^n$ depend on the parity of $n$ is another clue in this regard."

Clearly, each of $\pi$, $2\pi$, $\pi/2$, $\pi/4$ and $2\pi i$ have their benefits, but should we seriously isolate $2\pi$ and attempt to redefine it as $\tau$? Sure $\tau$ is better in a few instances, but that is because it is a multiple of $\pi$. This is no reason to introduce a new constant and encourage mathematics to adopt it.

The main argument for $\tau$ is its simplicity to calculate the number of radians in a fraction of a circle. I think we all would agree that $\tau$ makes this trivial task a bit more trivial. A tauist would ask you:

Quick, how many radians in an eighth of a circle?In terms of turns, $\tau$ has a slight advantage. Just look at the following two figures that appeared in the Tau Manifesto and tell me you aren't convinced by the power of $\tau$!

Is it $\pi/4$ or $\tau/8$?

Figure 1: Some common angles. (Source: tauday.com)

But this is not a reason to switch to $\tau$. The context is highly relevant in this regard and similar questions which favor $\pi$ can be posed. Let me demonstrate with an example by using areas rather than angles. Note that the area of a unit circle is $\pi$.

Now quick, what is the area of an eigth of a unit circle?Tau may have its benefits when looking at turns, but when looking at areas $\pi$ takes the cake (or rather, pie). Just like the Tau Manifesto, I too can create convincing looking pictures:

$\pi/8$ or $\tau/16$?

Figure 2: Areas of particular sectors of a unit circle.

Looking at Figure 2, it seems that $\tau$ is off by a factor of two. This demonstrates that in

As demonstrated in Section $3.1$, when dealing with areas $\pi$ outperforms $\tau$.
One of the most important problems proposed by ancient geometers is that of *squaring the circle*.
The problem is stated as:

Can a square with the same area as a circle be constructed by using only a finite number of steps with compass and straightedge?

Figure 3: Squaring the circle.

Reinterpretting this problem in terms of $\tau$ is a disaster, which provides more motivation why $\pi$ is the true circle constant. To conclude this section we reiterate a very important fact:

This result is so beautiful that it would be a crime to rewrite it using $\tau$.The area of a unit circle is $\pi$.

The Gaussian integral is the integral of the Gaussian function $e^{-x^2}$ over the entire real line: $$\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.$$ This integral is important and has many applications in mathematics. Notice the integral does not have a $2\pi$, beautiful!! This is when tauists will claim there is a similar formula with $2\pi$ in it, but then we end up with a nasty fraction of $1/2$ in the power of $e$, ew! The only thing worse than multiplying by $2$ is dividing by $2$: $$\int_{-\infty}^\infty e^{-x^2/2}\,dx = \sqrt{\tau}.$$ Comparing these two integrals most mathematicians would agree that not only is the first one nicer, it is much more natural! When the Gaussian integral is normalized so that its value is $1$, it is the density function of the normal distribution: $$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\, e^{\frac{-(x-\mu)^2}{2\sigma^2}}.$$ However, by grouping the $2$ with the $\sigma^2$ rather than with the $\pi$, it can easily be written in the form $$f(x)=\frac{1}{\sqrt\pi(\sqrt 2\sigma)}e^{\frac{-(x-\mu)^2}{(\sqrt 2\sigma)^2}}.$$ The Tau Manifesto groups the $2$ with the $\pi$ and gives this formula as an example where $\tau$ wins over $\pi$. But in fact, the $2$ does not belong with the $\pi$ and this becomes even more apparent when looking at alternate suggestions for the "standard" normal distribution. The distribution with $\mu=0$ and $\sigma^2=1$ is called the standard normal, that is, $$\phi(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{1}{2} x^2}.$$ Various mathematicians debate on what we should call the standard normal distribution. Note that above by setting $\sigma^2=1$ and grouping the $2$ with $\pi$ rather than the $\sigma^2$, it (falsely) appears to be a win for $\tau$. Gauss suggests that the standard normal should be $$f(x) = \frac{1}{\sqrt\pi}\,e^{-x^2}$$ and Stigler insists the standard normal to be $$f(x) = e^{-\pi x^2}.$$ Neither of these suggestions has a $2\pi$ because the $2$ does not belong with the $\pi$ in the first place. Unfortunately, $\phi(x)$ has been adopted as the standard normal, but this does not make it a win for $\tau$.

When analyzing other distributions we see that $2\pi$ is not as common in statistics as the Tau Manifesto would lead you to believe. The Cauchy distribution has the probability density function $$f(x)= { 1 \over \pi } \left[ { \gamma \over (x - x_0)^2 + \gamma^2 } \right],$$ and the standard Cauchy distribution has probability density function $$f(x)=\frac{1}{\pi(1+x^2)}.$$ The student's t-distribution has the probability density function $$f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{1}{2}(\nu+1)}.$$ Neither of these have a $2\pi$ appearing, but the student's t-distribution does have multiples of $\pi$ occuring. In fact, multiples of $\pi$ show up throughout mathematics, so it is no surprise that $2\pi$ shows up in some formulas.

Consider a triangle with interior angles $\alpha$, $\beta$ and $\gamma$. Let me ask you, what is the sum of these three angles? Is it $\tau$? That would be nice if it were, but in fact, the answer is the almighty $\pi$! $$\alpha+\beta+\gamma=\pi.$$ By looking at polygons we see that $\pi$ is a clear winner over $\tau$. Take any polygon with $k$-sides and interior angles $\theta_i$ (for $i=1,2,\ldots,k$). Then the sum of the angles is equal to $$\sum_{i=1}^k \theta_i=(k-2)\pi.$$ Once we look beyond specific angles inside of circles, $\pi$ really does show who's boss! In fact, multiples of $\pi$ are very important in mathematics, including $\tau=2\pi$. The importance of $\tau$ comes from the fact that it is a multiple of $\pi$, but other multiples of $\pi$ are just as important.

We have demonstrated that angles of arcs in circles are a win for $\tau$, interior angles in polygons are a win for $\pi$, areas in circles are a win for $\pi$, but what about areas of polygons? It is well known that the area of a regular $n$-gon inscribed in a unit circle is: $$A=n\sin\frac{\pi}{n}\cos\frac{\pi}{n}.$$ Clearly, another win for $\pi$.

We just can't stress this enough. The reason $\tau$ shows up a lot is because it is a multiple of $\pi$. We saw the multiple $\nu\pi$ appear in Section $4.2$ and the multiple $(k-2)\pi$ in Section $5$. Looking at trigonometric functions we should expect multiples of $\pi$ to again show up (and indeed they do). The following table shows the domain and period of common trig functions: $$\begin{array}{c|c|c} \mbox{Function} & \mbox{Domain} & \mbox{Period}\\ \hline \sin\theta & \mathbb{R} & 2\pi\\ \cos\theta & \mathbb{R} & 2\pi\\ \tan\theta & \theta\neq (n+\frac{1}{2})\pi,~~n\in\mathbb{Z} & \pi\\ \csc\theta & \theta\neq n\pi,~~n\in\mathbb{Z} & 2\pi\\ \sec\theta & \theta\neq (n+\frac{1}{2})\pi,~~n\in\mathbb{Z} & 2\pi\\ \cot\theta & \theta\neq n\pi,~~n\in\mathbb{Z} & \pi\\ \end{array}$$ Notice that $\pi$ shows up, along with $2\pi$ and $n\pi$. By converting the table to $\tau$ we would get even more nasty fractions than are already there.

One reason the Tau Manifesto is able to convert so many readers is because of their version of Euler's identity. They claim that $$e^{i\tau}=1$$ is more elegant than the formula $$e^{i\pi}+1=0,$$ but any mathematician can see this is total nonsense. Sure, there may be a nice formula that uses $\tau$, but that is because $\tau$ is a multiple of $\pi$. In reality, there is also a nice formula for the multiple $3\pi$, but that doesn't mean we should start worshipping $3\pi$. The fact is, their version of the formula may look nice but it is much weaker than the original. Consider the function $e^{ix}$. We ask the following important question:

What is the smallest positive solution $x$ so that $e^{ix}$ is an integer?The answer comes as no surprise, it is $\pi$. That is, $\pi$ is the smallest number that brings imaginary powers of $e$ back to the real line. This is why $\pi$ is more important than $\tau$.

Furthermore, the equation $e^{i\pi}=-1$ is a much stronger result than $e^{i\tau}=1$, and the $\tau$ equation comes trivially from the first equation by squaring both sides: $$\left(e^{i\pi}\right)^2=\left(-1\right)^2\quad\implies\quad e^{i\tau}=1.$$ When it comes to Euler's identity, $\tau$ just can't compete with the powers of the almighty $\pi$.

I do not have a background in engineering but it is important to consider the applications of $\pi$. Regarding the introduction of $\tau$ in the Tau Manifesto, Gareth Boyd writes:

Dr Hartl's theoretical background would seem to be on show here. He has forgotten about the practical application of mathematics - engineering. Tau is already one of the most important symbols in mechanical engineering as it denotes shear stress. Additionally the ratio of diameter to circumference is very important when we work with bars of material or pipes. We tend not to purchase these by the radius. Perhaps a little more thought and debate are required in this matter before we start a revolution.It should be mentioned that the Tau Manifesto does give a good argument for using the symbol $\tau$. However, we question whether the constant $\tau=2\pi$ is actually needed in mathematics.

The connection between the formula
$$A=\frac{1}{2}\tau r^2$$
and some quadratic forms in physics is certainly interesting, but the traditional formula
$$A=\pi r^2$$
is already a quadratic form preferred by mathematicians.
The Tau Manifesto would lead you to believe there is a $1/2$ missing by comparing it to a few physics formulas, but then you would be forgetting about the connection the formula has to circles.
One strong fact that was mentioned in Section $3.2$ is that the area of a unit circle is $\pi$.
This fact is what makes $\pi$ more important than $\tau$ as a large number of of problems in geometry deal with areas.
In my opinion, the only benefit $\tau$ seems to have is that it makes computing angles of arcs in a circle a bit more trivial.
When looking at areas, $\pi$ certainly shines.
Even when looking at areas where no circle seems to be apparent, $\pi$ shines through.
In particular, the last six formulas in Section 7 show that the area under the specified functions is **equal** to $\pi$ -- this is truly amazing.

I hope you enjoyed reading the Pi Manifesto. This article is meant to be a fun discussion of the importance of $\pi$ and why $\pi$ is the right circle constant afterall. It is a first draft and any additional arguments, mathematical facts, or strengthening of the above points in the defence of $\pi$ are more than appreciated! Please feel free to contact me if you have questions or comments.

None yet. If you have any improvements to the Pi Manifesto let me know!

*The Pi Manifesto*. Copyright © 2011 by MSC. Please feel free to share *The Pi Manifesto*, which is available under the Creative Commons Attribution 3.0 Unported License.

- Albert Eagle.
*Elliptic Functions as They Should Be*. Galloway and Porter, Cambridge, 1958. - Michael Hartl.
*The Tau Manifesto*. Available online at http://tauday.com. - Robert Palais.
*$\pi$ Is Wrong!*. The Mathematical Intelligencer, Volume 23, Number 3, 2001, pp. 7–8. Available online at http://www.math.utah.edu/~palais/pi.html. - Walter Rudin.
*Principles of Mathematical Analysis*. McGraw-Hill, 1976.