Pi is wrong! Long live Tau!

At one point or another, we’ve all had a feeling that something is not quite right in the world. It’s a huge relief, therefore, to discover someone else who shares your suspicion. (I’m also surprised that it’s taken me this long to stumble on this!)

It has always baffled me why we define \(\pi\) to be the ratio of the circumference of a circle to its diameter, when it should clearly be the ratio of the circumference to its radius. This would make \(\pi\) become the constant 6.2831853…, or 2 times the current definition of \(\pi\).

Why should we do this? And what effect would this have?

Well, for starters, this would remove an unnecessary factor of 2 from a vast number of equations in modern physics and engineering.

Most importantly, however, this would greatly improve the intuitive significance of \(\pi\) for students of math and physics. \(\pi\) is supposed to be the “circle constant,” a constant that embodies a very deep relationship between angles, radii, arc lengths, and periodic functions.

The definition of a circle is the set of points in a plane that are a certain distance (the radius) from the center. The circumference of the circle is the arc length that these points trace out. The circle constant, therefore, should be the ratio of the circumference to the radius.

To avoid confusion, we’ll use the symbol tau (\(\tau\)) to be our new circle constant (as advocated by Michael Hartl, from the Greek τόρνος, meaning “turn”), and make it equal to 6.283…, or \(2\pi\).

In high school trigonometry class, students are required to make the painful transition from degrees to radians. And what’s the definition of a radian? It’s the ratio of the length of an arc (a partial circumference) to its radius! Our intuition should tell us that the ratio of a full circumference to the radius should be the circle constant.

Instead, students are taught that a full rotation is \(2\pi\) radians, and that the sine and cosine functions have a period of \(2\pi\). This is intuitively clunky and fails to illustrate the true beauty of the circle constant that \(\pi\) is supposed to be. This is surely part of the reason that so many students fail to grasp these relationships and end up hating mathematics. A full rotation should be τ radians! The period of the sine and cosine functions should be \(\tau\)!

But… wouldn’t we have to rewrite all of our textbooks and scientific papers that make use of \(\pi\)?

Yes, we would. And, in doing so, we would make them much easier to understand! You can read the Tau Manifesto website to see examples of the beautiful simplifications that \(\tau\) would bring to mathematics, so I won’t repeat them here. You can also read the original opinion piece by Bob Palais that explores this subject.

It’s not particularly surprising that the ancient Greeks used the diameter of a circle (instead of the radius) in their definition of \(\pi\), since the diameter is easier to measure, and also because they couldn’t have foreseen the ubiquity of this constant in virtually all sciences.

However, it’s a little unfortunate that someone like Euler, Leibniz, or Bernoulli didn’t pave the way for redefining \(\pi\) to be 6.283…, thus missing the opportunity to simplify mathematics for generations to come.

Aside from all the aesthetic improvements this would bring, considering how vitally important it is for more of our high school students (and beyond) to understand and appreciate mathematics, we need all the “optimizations” we can get to make mathematics more palatable for them. This surely has to be an optimization to consider seriously!

From now on, I’m a firm believer in tauism! Are you?

Good and bad science, and faster-than-light neutrinos

The results from the OPERA experiment at CERN have caused a huge stir in the media over the last two weeks, and with good reason, since they claim to have measured the arrival of a neutrino beam 60 nanoseconds faster than light.

Before we go on, let’s calm down a bit. Even if these results are somehow confirmed, it wouldn’t “prove Einstein wrong,” or cause scientists to stop using General and Special Relativity on a day-to-day basis. If anything, it would show that Einstein’s theory is incomplete, but no one is disputing this in the first place.

Relativity (general and special) has been put through dozens of independent, precise, elaborate tests, and passed every single one with astonishing accuracy, which means that there’s definitely something fundamentally correct about Einstein’s theory. It shouldn’t be thought of as some kind of “sitting duck” theory, just waiting to be overthrown.

Understandably, the current consensus among the world’s physicists seems to be that there was a measurement error in the OPERA experiment, or that the experimenters neglected to integrate some subtle factor that accounts for the missing 60 ns. (For a wonderfully accessible introduction to the OPERA experiment, as well as particle physics in general, read Matt Strassler’s blog. For a more thorough discussion of possible mistakes, read Lubos Motl’s post on the subject. It’s also worthwhile to read the comments on those blogs.)

Perhaps the most convincing evidence against this experiment is that we have observed neutrino emissions from supernovae (specifically SN 1987A), and these neutrinos more-or-less coincided with our observation of visible light from the same supernova. If neutrinos are really faster than light, we should have observed the neutrinos many months before we observed the light. The only loophole in this argument would be if the OPERA effect is energy dependent, since the OPERA neutrinos had much more energy than the ones from the supernova, but that would present even more problems.

Not being a particle physicist myself, I can’t meaningfully contribute to the discussions on theoretical implications of this experiment, if it’s actually true. I would, however, like to comment on how this story is unfolding from the point of view of the scientific method, and specifically how this story highlights the differences between real science and pseudoscience. I use “pseudoscience” to refer to homeopathy, energy healing products, reiki, dowsing, magnets, pendulums, astrology, and anything else that requires more “faith” than evidence.

In the wake of attending a New Age expo (out of morbid curiosity) and being overloaded with crackpots, quacks, and hucksters, these differences become all the more plain:

  • The fact that the experimenters published any data at all is a sign of great scientific integrity. The fact that they held a press conference before the paper was peer-reviewed is a bit unfortunate, as noted by Lawrence Krauss, but I think the fact that this story made it to mainstream media outlets will help the general public understand the scientific process, as people follow the story. Pseudoscientists, on the other hand, seem to be allergic to data in general, and never publish anything.
  • Essentially, the scientists of the OPERA experiment are saying, “We’ve gathered these data, we used the best possible experimental parameters, we’ve performed all the checks we could think of, and we still see this anomaly. So please, tell us what we did wrong.” This is surely science at its best! This is the kind of behavior that should be an inspiration for a whole generation of new scientists. We will never hear pseudoscientists utter that phrase.
  • Real scientists don’t adhere dogmatically to any theory, no matter how foundational it may be. Even though most physicists agree that there was an error in the OPERA experiment, they still reserve a little room for the possibility that the results are correct, and that Relativity might be violated. Einstein to physicists is not the same as Chopra is to pseudoscientists.
  • Real scientists expect extraordinary evidence for extraordinary claims. Most scientists agree that the evidence collected by the OPERA experiment is not extraordinary. Pseudoscientists make extraordinary claims every time they open their mouth, but present no evidence at all, except anecdotal testimonials from their friends and paid endorsers.
  • If we read the blogs of popular physicists on the subject of the OPERA experiment, we find lively debates on theoretical explanations for the anomalous effect, and discussions on ways the experimenters miscalculated the speed of the neutrinos. The key point is: scientists get excited about the possibility of being proven wrong. Scientists can’t wait to be proven wrong, because it would mean that there’s more science to be done!
  • Perhaps most importantly, real scientists are motivated by a desire to better understand our world. The only motivation of pseudoscientists is money, thinly veiled by a scientific-sounding sales pitch, and a nonsensical product du jour.

In any case, I encourage everyone to follow this story, because it’s a high-profile example of real science at work; a triumph of human achievement. No matter how the results turn out, by observing the process of scientific scrutiny, everyone will be better equipped to spot pseudoscience when it’s in plain sight.

I will update this post as soon as I see a quack energy-healing product that uses faster-than-light neutrinos to balance the flow of energy through your chakras. Post a comment if you find one yourself!