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Why Tau Trumps Pi

A growing movement argues that killing pi would make mathematics simpler, easier and even more beautiful
pi vs. tau


Almost every mathematical equation about circles is written in terms of r for radius. Tau is precisely the number that connects a circumference to that quantity.
Credit: Dennis Wilkinson via Flickr

There aren't many things that Congress can agree on, but in early 2009 it passed a bipartisan resolution designating March 14th of each year as "Pi Day." Pi, the mathematical constant that students first encounter with the geometry of circles, equals about 3.14, hence its celebration on March 14. The math holiday had been a staple of geeks and teachers for years—festivities include eating pie the pastry while talking about pi the number—but dissent began to appear from an unexpected quarter: a vocal and growing minority of mathematicians who rally around the radical proposition that pi is wrong.

They don't mean anything has been miscalculated. Pi (π) still equals the same infinite string of never-repeating digits. Rather, according to The Tau Manifesto, "pi is a confusing and unnatural choice for the circle constant." Far more relevant, according to the algebraic apostates, is 2π, aka tau.

Manifesto author Michael Hartl received his PhD in theoretical physics from the California Institute of Technology and is only one in a string of established players beginning to question the orthodoxy. Last year the University of Oxford hosted a daylong conference titled "Tau versus Pi: Fixing a 250-Year-Old Mistake." In 2012 the Massachusetts Institute of Technology modified its practice of letting applicants know admissions decisions on Pi Day by further specifying that it will happen at tau time—that is, at 6:28 P.M. The Internet glommed onto the topic as well, with its traditional fervor for whimsical causes. YouTube videos on the subject abound with millions of views and feisty comment sections—hardly a common occurrence in mathematical debates.


The crux of the argument is that pi is a ratio comparing a circle’s circumference with its diameter, which is not a quantity mathematicians generally care about. In fact, almost every mathematical equation about circles is written in terms of r for radius. Tau is precisely the number that connects a circumference to that quantity.

But usage of pi extends far beyond the geometry of circles. Critical mathematical applications such as Fourier transforms, Riemann zeta functions, Gaussian distributions, roots of unity, integrating over polar coordinates and pretty much anything involving trigonometry employs pi. And throughout these diverse mathematical areas the constant π is preceded by the number 2 more often than not. Tauists (yes, they call themselves tauists) have compiled exhaustively long lists of equations—both common and esoteric, in both mathematics and physics—with 2π holding a central place. If 2π is the perennial theme, the almost magically recurring number across myriad branches of mathematics, shouldn’t that be the fundamental constant we name and celebrate?

If that’s all there was, the tau movement would likely be a curiosity and nothing more. But reasons for switching to tau are deeply rooted in pedagogy as well. University of Utah mathematics professor Robert Palais, who is considered the founding father of the movement, started the "pi is wrong" ruckus with an article of the same name in 2001[pdf]. The article, which should be required reading for all advanced high school students, creates a tantalizing picture of how much easier certain fundamental concepts of trigonometry could be in an alternate universe where we use tau. For example, with pi-based thinking, if you want to designate a point one third of the way around the circle, you say it has gone two thirds pi radians. Three quarters around the same circle has gone one and a half pi radians. Everything is distorted by a confusing factor of two. By contrast, a third of a circle is a third of tau. Three quarters of a circle is three quarters tau. As a result of pi, Palais says, "the opportunity to impress students with a beautiful and natural simplification is turned into an absurd exercise in memorization and dogma."

At its heart, pi refers to a semicircle, whereas tau refers to the circle in its entirety. Mathematician and poet Mike Keith once wrote a 10,000 word poem dedicated to the first 10,000 digits of pi. He is now a proponent of tau. According to a PBS article from last year, he said that thinking in terms of pi is like reaching your destination and saying you're twice halfway there.

For mathematicians, pi obscures some of the underlying symmetries of mathematics and muddies up what should be elegant with extraneous factors of two. There’s an admittedly grandiose idea that mathematics is the language with which we express and see certain underpinning truths to the universe. To clutter that language with superfluous twos would be as bad as littering a Shakespearean monologue with “likes” and “umms” and “whatevers.” As the Bard nearly wrote, “Knowledge is two of the half-wings wherewith we fly to heaven.”

We Americans have almost a proud tradition of using poorly chosen units because of inertia: Fahrenheit instead of Celsius, miles instead of kilometers. Even the great Benjamin Franklin inadvertently established the convention of calling positive charge negative and vice-versa as a result of his experiments with electricity.

Indeed, the whole problem began as a historical accident, tauists say. In early civilizations a diameter was an easier quantity to measure than a radius. So when the Babylonians or Egyptians wanted rules of thumb for their architecture, a ratio of circumference to diameter is what they turned to. (The two civilizations estimated it to be 3.125 and 3.16, respectively.) Even the Bible specifies the ratio of a circle’s diameter to its circumference: “And [Hiram] made a molten sea, 10 cubits from the one brim to the other: it was round all about, and…a line of thirty cubits did compass it round about” (1 Kings 7:23).

The Greeks used formal geometric proofs to estimate the circumference-to-diameter ratio. Archimedes (he of the lever and shouts of "Eureka!") found strict lower and upper bounds of 3.1408 and 3.1429. Yet his choice of comparing the circumference with diameter was arbitrary; he could just as easily have used radius instead. (Interestingly, Archimedes did not use the Greek letter π. That didn't come until Swiss mathematician Leonhard Euler popularized the convention in 1736, and even he seemed to be ambivalent about whether to define π as 3.14 or as the 6.28 we now write as τ.)

Although switching to tau when all the textbooks and academic papers use pi may sound daunting, it doesn’t need to be. There could be a transitional period of using both mathematical constants while we phase out the old and humor the intransigents who can’t or won’t change.

Asked in an e-mail about the reaction his original piece has received, Palais is humbled. "I never would have imagined the scale of the discussion," he says. And given that it's already far exceeded his expectations, he expresses optimism that it could continue even further.

Tau Day is approaching. It occurs, of course, on 6/28. As the Internet braces itself for the annual controversy, some lament the loss of a pun that embracing tau would entail. “But pie is yummy" remains one of the more compelling arguments for clinging to the traditional ways of 3.14. But tauists have a response for this as well: on Tau Day you get to eat twice as much pie!

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