Tuesday, May 5, 2015

Is pi really universal?

Full disclosure: I have the first 51 digits of pi tattooed on my left calf and foot, so I am personally and permanently attached to the traditional definition of pi.
My left leg
Pi is a fundamental constant of the universe, and it is defined to be a circle's circumference divided by its diameter, or the distance around a circle divided by the distance across it. The number pi shows up in math and physics on a regular basis, even when we are not talking about circles, and it is safe to say that without an understanding of pi, we would not be as technically advanced as we are as a species today.

But if NASA is right, we will discover alien life in the next 20 years, and while that life will probably just be microbial, as a species, we should start thinking about how we would communicate with intelligent life, should we ever find it. Math is a universal language, so it is only a matter of time before humans try to show off our intelligence by demonstrating our knowledge of pi, but... what if we are wrong?
The definition of pi involves some arbitrary choices. Why do we use the diameter instead of the radius? Why is it circumference divided by diameter instead of the other way around? The choices we make while defining fundamental constants like pi indicate our understanding of the universe, so we should think hard about whether our definition of pi is "right" before we go bragging about our mathematical prowess to newfound intelligent life.

I will take a look at a few possible definitions of pi here, but keep in mind this is by no means an exhaustive list. I am just trying to get you thinking about a number we usually take for granted.

Good old-fashioned pi, the way grandma used to calculate:

Pi as we know it today describes the ratio of the diameter to the circumference, and we have been using it for thousands of years. It must be a useful number in its present form for us to use it as-is for so many years, right?

Tau, the new approach to pi that makes trigonometry more intuitive

Tau is just like pi, but instead of using the diameter, we use the radius. This is a small change (tau is just pi times 2), but it does make a lot of trigonometric functions more intuitive. There are a lot of good arguments for tau, which you can see in a cute video here:
and you can even watch a talk from Michael Hartl, the author of the tao manifesto here:
The reason tau works so well is because when we do calculations with angles or polar coordinates, we use the radius of a circle and not the diameter. Using tau makes us consistent mathematicians.

Volume-centric pi, tied to the three spatial dimensions we live in

The reason tau adds elegance to certain calculations is because it focuses on the radius, and when we use pi in math and physics, it is often related to the concept of rotational symmetry. It comes down to the fact that there is no right answer to the question "which direction is forward?" You can spin your body around however you like, and decide for yourself which way to point. Pi (or tau) somehow represents this freedom to assign orientation arbitrarily.

The problem with pi and tau is that they are both "circle constants" and we live in a universe with three spacial dimensions. The fundamental principal this number describes could very well be the ability to assign an arbitrary direction in three dimensional space instead of just on a circle.

If we want to capture the three dimensional nature of our universe, then the "right" formulation of pi could look like the surface area of a sphere divided by the radius squared (4*pi), or the volume of a sphere divided by the radius cubed (4/3*pi).

We don't know the answer yet

We've looked at four possible formulations of this universal constant (pi, 2*pi, 4*pi, 4/3*pi). I'd love to tell you the "right" answer, but the truth is, pi is just a glimpse into a very profound and fundamental truth about the universe we live in, and just because we first noticed it while looking at circles does not mean it is a "circle constant." We need to understand why we live in a universe with exactly three spatial dimensions and rotational symmetry before we can try to postulate the "right" formulation. Perhaps this is a question another species out there can help us answer.