***THIS IS NOT OUR OWN WORK***## The Pi Manifesto

## Written by MSC

## [email protected]

## Last updated July 4th, 2011

## 1 π versus τ

## 1.1 The Tau Movement

*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
*π 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 π is the wrong circle constant and believe the true circle constant should be τ=2π. They celebrate Tau Day (June 28th), wear τ-shirts and spread pro-tau propoganda.

**But are tauists doing more harm than good?**

In this article we will explore this very question and provide several reasons why π will prevail in the intriguing π versus τ battle.

## 1.2 Any publicity is good publicity

*π 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 2πwith τ in serious mathematical circles."

"Either one is just fine, it won't make any difference to mathematics."

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

"Of course it had to be a physicist who would want to get rid of the usage of π... Theconceptof π 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."

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 τ in the defence of π. Any suggestions and reasons why π is better than τ (or τ is better than π) are more than welcome!

## 1.3 The Tau Manifesto is wrong

## 2 Definitions of π

## 2.1 The Traditional Definition

\[\pi\equiv\frac{C}{D}\approx 3.14159... \]

The manifesto then goes on to suggest that we should be more focused on the ratio of a circle's circumference to its radius:

\[\tau\equiv\frac{C}{r}\approx 6.283185...\]

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 $\displaystyle \frac{C}{R}$ 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, π 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 $\displaystyle \frac{C}{D}$ . 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)

## 2.2 Other definitions of π

Another common geometric definition for π is in terms of areas rather than lengths. Take r to be the radius of a circle. Define π 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 τ, this definition is messy and includes a factor of 2. In particular, define τ 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 π over τ and also involves the

*important*radius of a circle. Like the traditional definition, this definition of π depends on results of Euclidean geometry and comes naturally when looking at areas.

## 2.3 Why stop at redefining π?

\[\frac{\pi}{4}\equiv\frac{A}{D^2}.\]

This suggests that perhaps both π and τ are wrong, and $\displaystyle \frac{\pi}{4}$ is the correct circle constant. Others have also suggested similar numbers as the circle constant. In 1958, Eagle suggests that $\displaystyle \frac{\pi}{2}$ is the correct circle constant [1]. In fact, the $\displaystyle \frac{\pi}{2}$ Manifesto is coming soon to a website near you! (Just kidding, I hope). But why stop at redefining π? Terry Tao says:

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

## 3 Silly arguments

## 3.1 A silly argument for τ

Quick, how many radians in an eighth of a circle?

Is it $\displaystyle \frac{\pi}{4}$ or $\displaystyle \frac{\tau}{8}$ ?

Now quick, what is the area of an eigth of a unit circle?

$\displaystyle \frac{\pi}{8}$ or $\displaystyle \frac{\tau}{16}$ ?

*some*situations τ may be better, and in other situations π is better. The reason the Tau Manifesto is so convincing is because of selection bias. They only demonstrate the situations where τ is either better than π or comparable and ignore situations where it is worse.

## 3.2 A silly argument for π

*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?

The area of a unit circle is π.

## 4 Probability and statistics - A win for π

## 4.1 The normal distribution

\[\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.\]

This integral is important and has many applications in mathematics. Notice the integral does not have a 2π, beautiful!! This is when tauists will claim there is a similar formula with 2π in it, but then we end up with a nasty fraction of $\displaystyle \frac{1}{2}$ 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 $\displaystyle \sigma^2$ rather than with the π, 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 π and gives this formula as an example where τ wins over π. But in fact, the 2 does not belong with the π and this becomes even more apparent when looking at alternate suggestions for the "standard" normal distribution. The distribution with μ=0 and $\displaystyle \sigma^2=1$ 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 $\displaystyle \sigma^2=1$ and grouping the 2 with π rather than the $\displaystyle \sigma^2$ it (falsely) appears to be a win for τ. 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 2π because the 2 does not belong with the π in the first place. Unfortunately, ϕ(x) has been adopted as the standard normal, but this does not make it a win for τ.

## 4.2 Other distributions

$$f(x)= { 1 \over \pi } \left[ { \gamma \over (x - x_0)^2 + \gamma^2 } \right],$$

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 2π appearing, but the student's t-distribution does have multiples of πoccuring. In fact, multiples of π show up throughout mathematics, so it is no surprise that 2π shows up in some formulas.

## 5 Polygons and triangles - Another win for π

$$\alpha+\beta+\gamma=\pi.$$

By looking at polygons we see that π is a clear winner over τ. Take any polygon with k-sides and interior angles $\theta_i$ (for i=1,2,…,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, π really does show who's boss! In fact, multiples of π are very important in mathematics, including τ=2π. The importance of τ comes from the fact that it is a multiple of π, but other multiples of π are just as important.

We have demonstrated that angles of arcs in circles are a win for τ, interior angles in polygons are a win for π, areas in circles are a win for π, 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 π.

## 6 Trigonometric 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 π shows up, along with 2π and nπ. By converting the table to τ we would get even more nasty fractions than are already there.

## 7 Other formulas

The error function:

$$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt.$$

The sinc function:

$$\operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}.$$

The Gamma function:

$$\Gamma(1/2)=\sqrt\pi$$$$\Gamma(3/2)=\sqrt\pi/2$$$$\Gamma(5/2)=3\sqrt\pi/4$$

Euler's Reflection Formula:

$$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}.$$

Volume of unit n-ball:

$$V=\frac{\sqrt\pi^n}{\Gamma(1+\frac{n}{2})}.$$

Area of an ellipse:

$$A=\pi ab$$

Integral of hyperbolic secant:

$$\int_{-\infty}^\infty \operatorname{sech}(x)\,dx = \pi.$$

Integral of $\displaystyle \frac{1}{\sqrt{1-x^2}}$:

$$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\,dx = \pi.$$

Integral of $\displaystyle \frac{1-\cos x}{x^2}$:

$$\int_{-\infty}^\infty \frac{1-\cos x}{x^2}\,dx = \pi.$$

Integral of $\displaystyle \frac{\sin x}{x}$:

$$\int_{-\infty}^\infty \frac{\sin x}{x}\,dx = \pi.$$

Integral of $\displaystyle \frac{\sin^2 x}{x^2}$:

$$\int_{-\infty}^\infty \frac{\sin^2 x}{x^2}\,dx = \pi.$$

Integral of $\displaystyle \frac{1}{1+x^2}$:

\[\int_{-\infty}^\infty \frac{1}{1+x^2}\,dx = \pi.\]

Where are you τ? Ah, it must be hiding in shame.

## 8 Euler's Identity

$$e^{i\tau}=1$$

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 τ, but that is because τ is a multiple of π. In reality, there is also a nice formula for the multiple 3π, but that doesn't mean we should start worshipping 3π. The fact is, their version of the formula may look nice but it is much weaker than the original. Consider the function eix. We ask the following important question:

What is the smallest positive solution x so that $\displaystyle e^{ix}$ is an integer?

Furthermore, the equation $\displaystyle e^{i\pi}=1-$ is a much stronger result than $\displaystyle e^{i\tau}=1$ and the τ 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, τ just can't compete with the powers of the almighty π.

## 9 Conclusion and remarks

## 9.1 Engineers are against τ

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.

## 9.2 Quadratic Forms

$$A=\frac{1}{2}\tau r^2$$

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 $\displaystyle \frac{1}{2}$ 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 π. This fact is what makes π more important than τ as a large number of of problems in geometry deal with areas. In my opinion, the only benefit τ seems to have is that it makes computing angles of arcs in a circle a bit more trivial. When looking at areas, π certainly shines. Even when looking at areas where no circle seems to be apparent, π shines through. In particular, the last six formulas in Section 7 show that the area under the specified functions is

**equal**to π -- this is truly amazing.

## 9.3 From the author

## Acknowledgments

## Copyright and license

*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.
*π 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.