# The Tau of Pi

Pi Day. A day to celebrate that transcendental number, π. A day to eat circular desserts, and reflect on the nature of the most famous constant in mathematics.

No matter what field of science, technology, engineering, or mathematics you may be in, π probably plays a prominent part. Yet certain tau infidels dispute this.

## I Like Pi!

π, sometimes called pi, is possibly the most famous mathematical constant of all. It’s defined as the ratio of the circumference of a circle to its diameter:

$\pi = \frac{C}{d} \approx 3.1415926535897932$

This number shows up all sorts of places, yet the heathen Tauists believe that the true circle constant should be the ratio of the circumference of a circle to its radius:

$\tau = \frac{C}{r} \approx 6.28$

Why?

## How Great Tau Art?

Advocates of τ make a few points, the first typically being the definition of a radian. The radian is a unit of angular measure, defined such that an angle of 1 radian is measured by a circular arc whose length is equal to the radius of the circle.

Based on that definition, there are 2π radians in a complete circle. A Tauist would point out that it looks more natural for there to just be τ radians in a complete circle. In fact, they suggest that τ is not a near-lookalike of π, but actually stands for “turning”.

It’s understandable that these Tauists prefer that τ radians = 360°. This is a common affliction among physicists as well, who create systems of measure that set various physical properties equal to 1. Simple multiplication can be quite an annoyance, after all.

Incidentally, τ already has multiple meanings in physics and engineering, including torque (τ), sheer stress (τ), and the tau (τ) lepton. It has various other meanings elsewhere. Think this doesn’t matter?

Here’s a little imaginary tale.

It involves the imaginary number, $i = \sqrt{-1}$

Despite being imaginary, this number appears quite often in STEM fields. Yes, even in engineering, which usually concerns itself with “real” values.

Except in electrical engineering.

You see, the symbol $i$ was already in use in electrical engineering by the time that complex analysis worked its way into the field. It denotes a time-varying current, as opposed to $I$ which denotes a constant current. (This nomenclature is consistent elsewhere as well: $v$ is time-varying voltage, whereas $V$ is DC.)

So what did we do? We called the imaginary number $j$ instead of $i$.

$F(\omega) = \int f(t) e^{j \omega t} dt$.

It looks funny, and it gives us “imaginary number $j$“, but it lets us keep our time-varying current $i$. If a symbol is already in common use in a field, the field will not change its meaning.

## Far as the Pi Can See

Although τ shows up in a few mathematical equations, π shows up in numerous places in statistics and in engineering, including many places where replacement with τ would introduce an unnatural factor of $\frac{1}{2}$.

Kidding aside, I have nothing against tau. It’s sometimes useful to look at things from a new perspective.

In Carl Sagan’s Contact, the SETI researchers search for alien radio signals at a frequency of $\text{Hydrogen} \times \pi \approx 4.4623 \text{GHz}$. If the aliens had been Tauists, Dr. Arroway would have missed their radio signal.

(On the other hand, the fact that Dr. Arroway later finds interesting structure in the digits of π suggests that God Himself prefers π to τ.)

Why did I bring up Carl Sagan? Maybe my mind is just on billions and billions of pies. Mmmm… pie.

So I’ll close with one of the most famous equations in mathematics: Euler’s Identity. It’s been called the most beautiful equation in mathematics, because it relates five important mathematical constants: e, i, π, 1, and 0.

Marvel in its magnificent mathematical beauty. Happy Pi Day!

$e^{i \pi} + 1 = 0$