Pi vs. Tau vs. Eta

Published on Thursday, June 28, 2012 in , , , , ,

Michael Hartl's Tau-based unit circleToday is 6/28, making it Tau Day! Tau is a constant equal to 2 * Pi.

It sounds simple, and suggests that Tau shouldn't be a big deal. However, there is an entire movement to have Tau take Pi's place!

Let's start with good old familiar Pi (π). The following video, The Story of Pi, is a great refresher course on a very useful constant:

One of the most useful places for Pi is throughout the unit circle. If you need a refresher, here's a clear and simple Unit Circle tutorial.

With those concepts in mind, let's have Vi Hart explain why Tau (τ) is a better alternative to Pi.

There's plenty of support for Tau. Dr. Kevin Houston has another excellent intro to Tau, and Vi Hart even recently released a Tau Day song.

Michael Hartl, who began the Tau movement, offers a much deeper look at the effects of Tau in a 51-minute lecture, as part of his full Tau Day website. The Tau Manifesto itself is also available on the site for free.

Bob Palais classic article π is Wrong! (PDF link) is also considered to be a must-read for Tau enthusiasts.

Much of that was around for Tau Day 2011, however. This year, however, Tau faces an entirely new challenge!

The new challenger is Eta (η), which is ½π, effectively the inverse of Tau. David Butler's video shows the surprising uses of Eta, some of which rival Tau AND Pi!

That only leaves us π Day being on March 14th (3/14) and τ Day being on June 28 (6/28) is well and good, but when would we celebrate η Day? January 57th? Perhaps February 26th?

Why are we looking solely at the constant for change? How about changing the measuring unit? Since π is defined as a circle's circumference to its diameter, why not measure the circle in diameter lengths instead of radius lengths?

We could call these units “diameterians”, which would be equal to 2 radians. A complete circle would have π diameterians.

Notice that even from the seemingly simple shape of a circle, we can look at it from so many different ways. That's been part of the fun of putting up this post.

What do you think is the way to go? π? τ? η Diameterians? I'd love to hear your thoughts in the comments!

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3 Response to Pi vs. Tau vs. Eta

10:39 AM

Pi. This whole pi vs. tau thing is nonsense and eta is just adding fuel to the fire. It's really not a big deal. I also love how pi reflects the long history of mathematics. We use circumference/diameter and not circumference/radius because with primitive tools it was so much easier to measure the diameter of a circle than the radius. These roots are everywhere. Like geometry, which literally means 'land measurement,' shows how today's complex math comes from Greek farmers trying to plot out fields efficiently. Keep it pi.

10:38 AM

I find this discussion to be utterly fascinating! I think everyone is right! Obviously, as a practical matter, for historical reasons as well as intellectual inertia and with hints of the qwerty phenomenon at work, pi is the clear winner and will be with us for quite a long time to come. After all, although the Good Lord may not play dice, God definitely ratified the choice of pi by having Albert Einstein himself born on Pi Day, March 14, 1879, a Friday! Vi Hart, the wonderful and precocious niece of one of my best friends Kathleen, makes a very persuasive case for tau = 2*pi and there are, indeed, far too many fundamentally important equations in physics and math that are littered with ungainly factors of 2*pi--for example, Euler's beautiful formula for the Euler-Riemann zeta function for any positive even number 2*n: zeta(2*n) = [(-1)^(n+1)]*{B_2*n}*[(2*pi)^(2*n)]/2/[(2*n)!], where B_2*n is a Bernoulli number! However, as a purely logical matter, I have to admit that David Butler's arguments for eta = pi/2 are the most persuasive! Personally, given my last name (Furlong), which is defined to be 1/8 of a mile and is most often encountered in horse racing, I think that kappa = tau/8 = pi/4 = eta/2 = 0.7853398163397448... would be another fine choice for a fundamental transcendental constant number if for no other reason than that kappa is the first Greek letter in the Greek word for a circle, kuklos! Upon Latinization, the Greek upsilon becomes the English "y" so that the etymology of "cycle" from kuklos is readily seen! Also, kappa is usefully a real number betweem 0 and 1, for whatever that's worth! Trigonometrically, sin(kappa) = cos(kappa), uniquely, and kappa is the unique angle any diagonal of a square makes with the sides of the square! Notwithstanding all the enchanting and intriguing arguments for tau and eta and kappa, I stay with the observation I started with, namely, that pi is here to stay supreme for quite a while, which makes my successful efforts at committing the first 1512 decimal digits of pi to memory seem slightly less ridiculous than they were! Why did I stop at 1512? Well, 1511 is the 240th prime number, there are 24 hours in a day, and 10 is the well-known and well-beloved base of our decimal system, and I couldn't properly round pi to 1511 decimal places without knowing the value of the 1512th decimal digit (which happens to be a 7, coincidentally the lucky number of days in a week, which causes the 1511th decimal digit, 8, to round up to 9, which also happens to be the 1510th decimal digit!)--the real reason is that I eventually became somewhat bored with the whole project, which was taking up too much time every day since I tended to remember all 1512 numbers better if I recited them in my head in conveniently bite-sized groups of 10 daily! Clearly, I also desperately needed to get a life!

1:03 PM

Definitely Tau. Because the number in front of Tau is equivalent to the portion of a circle traversed, it makes the geometrical meaning of equations so much clearer. E.g. A=1/2 tau r^2 might seem less clear but even there in the equation is a hint at the proof. Likewise e^i tau = 1 makes sense because an entire turn leaves you where you are. But it'll probably never change, so pi'll have to do