Two Pi or not Two Pi

The date I’m writing this blog post is March 14th, 2015. If you live in a very particular part of the world, you might represent that date as 3/14/15, which in turn might make you excited to see the first 5 digits of π all nicely lined up in a row. The rest of the world might be confused why you’re making such a big deal out of 14/03/15, and the Western engineers are just biding their time until it’s 3/14/16. The arbitrariness of π day not-withstanding, I’m here to talk about how π, and any days relating to it, pales in comparison to Tau (and any days relating to it!)

Still looks delicious, however. Image: Public domain, via Wikimedia Commons.

But first, some history: π, as we all should know, is the ratio between the diameter of a circle and its circumference. This is 3.141592…  for each and every circle EVER. Which is pretty cool! This usually marked an important discovery for each. Most early cultures didn’t get it quite right, but made their best rational number approximation. In ~1700 BCE, a Babylonian clay tablet uses a constant represented as 25/8, or 3.1250. Around that same time period, an Egyptian papyrus scroll approximates π as (16/9)^2, or 3.1605. One of the most successful techniques of approximating π during this time period is often attributed to Archimedes, where he calculates the perimeter of an inscribed N-gon. Archimedes used an 96-sided polygon, and calculated π to be somewhere between 23371<< 227 (roughly 13.1408 and 3.1429). At the same time, a famous Ancient Chinese mathematician, Liu Hui did the same algorithm on an 3,072 sided polygon, approximating π to be 157/50 3.1416. However, Liu Hui developed a similar, faster algorithm after noting that successive inscribed polygons formed a geometric sequence with a factor of four. Some 200 years later, another Chinese mathematician used this algorithm on a 12,288-sided polygon, calculating the π approximation out to be 31464625<100<31464625 +169625, which translates to roughly 3.141592920.

But enough about the history of π. What I couldn’t find, in all my research, is why ancient peoples were obsessed with the ratio of the circumference and diameter, rather than the circumference and the radius!? After all, a circle is literally defined by its radius as a distance from its origin. Why go through all that unnecessary doubling! I guess ancients needed their line segments to touch something ‘tangible’ on both sides, but thanks to them, let’s take a look at what we have to deal with:

The number π is such an entrenched constant that we developed a unit of measure to use it: radians (which are, of course, dimensionless). If we wanted to measure a full circle it is 2π radians. Wait, what? We have got this glorious constant that people constructed 12,000 sided polygons to calculate, and it only gets us halfway around the circle? So now π/2, π/4, π/8 don’t actually mean half of a circle, or a quarter of a circle, but half of a half of a circle, and quarter of a half of a circle. Enter Tau.

Why not use Tau to express the ratio of the circumference and the radius, that ever pivotal piece of circular information. Because the radius is exactly half the diameter, you can clearly see how Tau is 2π, or 6.28(ish). Why the letter Tau, though? Well my theory is that it is similar enough to Pi that folks don’t feel too threatened by it’s emergence. So what does Tau get us?

For starters, how many radians is a full circle using Tau? Just, Tau! And how about half a circle: Tau / 2! There is no need for a quick mental check of divide by two conversion (…or, wait, was it multiply by two? Which way are we going again? See how this is confusing!), what you want and what you’re looking for are simply in the constant you use.

See how much easier tau is? Image: Michael Hartl, via Wikimedia Commons.

“But Ryan,” you exclaim, “What about ei? Without π, how will Euler be identified!” And to that I say: never fear. While it is true that ei=-1, if you substitute in Tau for Euler’s Identity, you’ll find that cos() + isin() = 1 (Euler’s Formula substituting=) reduces down to a very tasty 1. Identity saved.

And the list goes on! A sine wave is still a sine wave, using instead of π. Using a constant derived from the actual construction of a circle, rather than a near arbitrary doubling of the radius that has been passed down from Ancient peoples and entrenched into our present day mathematical dogma, just seems to make more sense.

And best yet, our deliciou Pi(e) day of March 14th is only delayed a few short months until June 28th,which can be represented as 6/28 which as someone pointed out to me happen to be perfect numbers. This switch practically sells itself, to be honest…though we’ll have to come  up with a dessert called “Tau.”

Of course, there are alternative solutions…

Image: xkcd by Randall Munroe.

Sources:

http://en.wikipedia.org/wiki/Pi

http://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/

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